There are some Christians think you can settle such things by using quotes from the Bible and find passages that show reincarnation is not possible. But other Christians think that the Bible though it has inspiration in it is also work of human hands too, and have a more flexible approach to it.

The Bible also is often inconsistent with itself. E.g. the four accounts in the gospels are not consistent with each other. So you can ask again, "which is right". Most Christians don't think that is an issue.

And there are esoteric Christian movements that have ideas of reincarnation that are central to them, so they would be familiar with all the passages in the Bible used by those who say reincarnation is incompatible.

So I think you can't say anything here for "all Christians" but have to say "which particular group of Christians".

Theosophists particularly adopt their own version of reincarnation Theosophical Society in America, and Reincarnation - the evidence, also wikipedia article on Esoteric Christianity. One of their distinctive ideas is that a human being can only be reincarnated as another human being in the future.

A bit more on proofs from the scriptures: Some Christians think every word of the scriptures is divinely inspired. Some think even that particular translation into English is inspired, there are some who think that the King James bible translation is divinely inspired The King James Bible Is Inspired. So then you'd typically settle a question like this using quotes from the Bible.

Others say that there was scope for mistakes, the authors were only human, and point to e.g. discrepancies in the Bible (e.g. inconsistent descriptions of the same events depending on which gospel you read, and the New Testament contradicting passages in the Old Testament etc) and also say that to some extent what we have is a matter of editorial decision by later compilers - e.g. that there were many books left out of the "final edition" of the Bible, as well as a few inspiring passages and stories added to the text many centuries after Jesus. For instance the story of Jesus and the woman taken in adultery, just about everyone agrees is a much later addition to St John's gospel. But that doesn't stop it from being inspiring and a powerful Christian message.

So it would depend on your tradition of Christianity, and its interpretation of the Bible and what exactly it treats as divinely inspired and not capable of being questioned (if you belong to a tradition that says certain things can't be questioned) whether reincarnation is consistent or not.

As a Buddhist myself, though brought up as a Christian, I'm probably not the right person to go into that any more, into detailed for and against scriptural and other theological arguments.

But perhaps I can say a bit more, helpfully, about ideas of reincarnation.

There are many ideas about reincarnation - it is not a single theory or concept. The Theosophical ideas are very different from Hindu ideas, and those in turn are very different from Buddhist ideas.

And Jains are very different again. I think they are the only ones that talk extensively about the possibility of reincarnation as plants, and they say also that a tree can become enlightened apparently - and try to avoid harm to plants as far as possible not just to insects, animals and humans - though some Hindu texts go into this as well apparently, and also some Japanese Buddhists touch on that possibility.  For more details on the Jains, also some Hindu and Buddhist traditions and ideas of reincarnation as plants, and the extent to which they extend ideas of non harming to plants, see Plants as Persons

For most Buddhists at least, you can only be reincarnated as animals, of the creatures we know. For theosophists you can only be reincarnated as human.

Within Buddhism (which I know most about), there's a wide range of ideas also.

For instance, Tibetans think you take rebirth after a time of some weeks spent in the Bardo, intermediate state. And I think they are the only ones that actually sometimes identify previous rebirths. Buddha actually warned that it can be unproductive to try to find out who or what you were in a previous life. So many Buddhists don't think it is worth trying to find out. But in Tibet then in cases of some noted teachers then rebirths are identified.

Therevadhan Buddhists, e.g. from Sri Lanka think you take rebirth right away immediately after your previous death.

In Japan, then the Zen Buddhists believe in reincarnation in a general way but don't put much emphasis on it at all, don't have elaborate ideas and theories about how it happens.

Generally you just need an open mind about what happens when you die in Buddhism, there is no creed, like you have to believe this or that idea about reincarnation to be a Buddhist. Instead if you follow the path of the Buddha - then all your beliefs are open to question and look at.

Since other answers here seem to be based on Western ideas of reincarnation that often have little resemblance to the ideas as understood in the East, worth mentioning some misconceptions about rebirth - at least as understood in Buddhism - and much of this is also true of other Eastern religions.

1. You don't expect to remember your previous life. After all I don't remember anything from my very early childhood and most don't. Why expect to remember a previous life - which also has a previous death since then, loss of that previous body, rebirth from a foetus in this life? After all those experiences, you wouldn't expect to remember anything. People lose their past memories sometimes even just because of accidents. It's rather astonishing that some seem to remember previous lives.

Traditionally in Tibet for instance they say for the teachers whose reincarnations are recognized - that in their next rebirth as young children, they often recognize implements they used (e.g. a bell) from a previous life - and that can be part of the method used to find them. But that is only when they are young, they forget this as they grow older.

Most don't remember anything at all and I think right to be skeptical about claims of some people to remember past lives in elaborate detail - perhaps it can happen but humans can also get false memories rather easily that seem to you as if they actually happened in the past.

2. There is no idea that your rebirth will be in some sense the same person with the same likes and attitudes and views. How much of your personality and likes and beliefs and attitude to life do you share with the young child of 2, or the old person of 100 if you live that long? Even in this life, one can have many changes. Even as an adult. Some things interest me now that were of zero interest to me, say, forty years ago, and vice versa.

Famous examples, Dalai Lamas are very different in personality. Current DL is scholarly, thorough understanding of the Tibetan Buddhist scriptures and commentaries as well as more esoteric teachings in all four of the main Tibetan Buddhist schools.

Previous DLs include the famous sixth, a renowned poet who renounced his monks robes (which you can do) and wrote famous (in Tibet) love poems. His poetry is still loved by Tibetans today.

I've never heard of the current Dalai Lama writing poetry - if he does - it's not what he is noted for. (Some Tibetan Lamas are noted for their poetry, even writing in English, Trungpa Rinpoche wrote many poems in English but the Dalai Lama - I don't think that is his thing particularly).

3. Not at all the idea that it's the same atoms. Just continuity of awareness. Even in this life your body probably has very few actual atoms in common with that 2 year old child.

4. If you experience good fortune now, it doesn't mean you are better than someone who is having a bad time. Or even someone who was a mass murderer earlier in this lifetime.

Just matters where you are at now. As an example, the sutras have an example of a serial killer who became a disciple of the Buddha, regretted his past actions and became enlightened in that very lifetime as an example of how you can turn your life around. Many other examples.

So people who suffer - they did awful things in the past. But so did I, over countless past lives. So anything like that is something I'll experience too, if not in this life, in a future life, until I reach enlightenment.

But as well as that, in Buddhist teaching, is more like a vulnerability - through the negative past actions my body is vulnerable to pain and suffering. We can see that all humans are similarly vulnerable even if some are currently experiencing more fortune than others. And the ultimate cause of this, it's taught, is confusion and ignorance, not evil.

An example of how karma works given by a great contemporary Buddhist scholar Prayudh Payutto is that if you walk up a flight of stairs - you can no longer touch the ground, you may be out of breath, and you have a better view. All of those and many more things follow from your decision to climb the stairs.

It doesn't make you any less empathic to recognize that your actions in this world have consequences for yourself and others, and is the same over previous lives. If anything it makes you feel more in common with others, even from very different backgrounds. Their lives are all examples of situations I've been in in my own past and future lives, and vice versa.

5. The argument about numbers of humans is just like arguing that human population of New York increases in daytime and asking where they come from. Just means there is more to the universe than just humans. In Buddhist teaching, you could be animals in past lives but also many other kinds of beings, maybe living in another world system also, as we'd think of it, around another star. Or beings we don't know exist right here.

6. It's an ancient scholarly religion, not a superstition. Much study and thought and debate.

This is one of the oldest Buddhist universities, from fifth century AD. It was destroyed in the twelfth century by Muslim invaders

Nalanda

Which is also the reason India stopped being a Buddhist country - the Buddhist teachings by then had spread throughout the Eastern world with this one of the great Buddhist centers of learning.

Though not so well known in the West, Nalanda university is as famous in Buddhism and as much a center of learning as Plato's academy or the great library of Alexandria.

Scholarship goes back to long before then

Here is one of the early Buddhist philosophers in the Indian traditions

Nagarjuna

Here is one of several early scholar monks in the Chinese tradition
Lokaksema (Buddhist monk)

This continues to this day,

Walpola Rahula, a recent noted Sri Lankan Buddhist scholar in the Therevadhan Buddhist traditions.

And this is the Dalai Lama taking the exams for his Geshe degree - glimpses of old Tibet.

This degree requires many years of advanced study in Buddhist topics and he passed with flying colours astounding his examiners with his brilliance, so it is said.

I don't think many Westerners realize that our current Dalai Lama is considered by the Tibetan Buddhists to be especially erudite in his understanding of Tibetan Buddhism. He is very modest about it, you'd never know from his interviews on video etc.

7. Are many reasons for thinking reincarnation is likely. For me, it made a lot more sense of many things that I found very hard to accept and understand as a Christian such as the "problem of evil".

In Buddhism that's just result of past confusion like when you have a car crash through a poor decision. Doesn't mean you are intrinsically a bad person.

That makes a lot more sense to me than Christian ideas that it's the doings of an all powerful God acting in ways mysterious to us that has this effect that some people's lives are a misery and others very happy. I could never make a lot of sense of that as a Christian.

But I can accept that for Christians the idea that it is God who is responsible for this, for creating a world like this with all its imperfections and challenges, gives meaning to their lives especially when combined with ideas of redemption. So would never say they are wrong here. And Christian thinkers think long and hard about the "problem of evil" and have written many profound works about it.

It is just that Buddhism is a better fit for me and the path I chose to follow. And for us, it is not a problem of evil, but a problem of confusion :). And trying to find our way out of confusion and ignorance and short term thinking and such like.

8,, So, none of this is any reason for Christians to believe in reincarnation.

Just saying, a large part of the world's population does believe in reincarnation. But there are many others with other belief systems.

Is reasonably obvious that ones religion and belief system largely depends on where you were born and your culture. If you are brought up in a particular culture, and never encounter any other religion or culture, most people will share the religion and belief systems of that culture. It's very unlikely that you'd come to new ideas or start a new religion.

9, Extremist sects and other problems don't show that a religion is itself responsible.

There are extremist sects in all religions and also in non religious belief systems such as communism. For instance even in Christianity you have various groups involved in Christian terrorism which most Christians would say are not following the genuine path of Jesus.

On the whole, we are lucky to live in a world where all the main religions are ethical in their core teachings, emphasizing respect for others, tolerance, compassion, love, generosity, helping the disadvantaged etc.

That's true for instance of Christianity, Judaism, Islam, Hinduism, Jains, Buddhism, Taoism, Shintoism, and many of the religions based on ancestor worship and Shamanistic religions. (Not trying to give a comprehensive list here).

(for more on this see comment)

9. They all deserve respect I think, as much as Christianity does.

Respect which doesn't mean at all that one should try to mush the various religions together into a single system of thought.

For some, Christianity is the path and it works fine for them without reincarnation. Since there is no way to prove or disprove this why feel that it needs to be changed?

For others, then they believe in reincarnation while Christian, again what is the problem there?

It's not as if we can prove reincarnation, or Christian ideas of a life everlasting after death, or prove existence of the ancestors for ancestor worship etc etc. Nor can we prove that when you die that's the end of everything either. Scientists have far too little understanding of what mind is to attempt to prove far reaching things like that.

And - they may all be giving us insights, in different ways, into the human situation. But many of them are incompatible belief systems for people with our need to put things into tidy categories. So it works best, for most, to stay within one system while learning from others.

Some people with very flexible minds can see to the essence as it were and have no trouble practicing simultaneously in two different systems that to others seem incompatible.

See also Robert Walker's answer to Are some Buddhists also Christians?

But for many it helps to find your particular path which may well be within one or other of these great belief systems. Others find a path in things that to many seem mundane, e.g. you may find all the answers and path you need as a gardener or some such.