Yes, though it needs a very flexible mind. Elaine MacInnes is a Catholic nun and missionary, and also a Zen roshi or Zen master. See Nun draws on Zen Buddhism to enrich Catholicism

It doesn’t matter that the teachings are incompatible. If you have that flexibility of mind.

The Rime teachings of Tibetan Buddhism may help. It’s a half hour talk but Ringu Tulku talks here about how Tibetans learnt to deal with the situation that they have several mutually incompatible branches of Buddhism - they contradict each other just as much as e.g. Catholic teachings contradict Protestant teachings in Christianity. Could you be simultaneously Quaker and Catholic for instance? Well yes, you could if you have that flexibility of mind though for most people it would be very confusing to whole heartedly follow both at once.

Tibetans with a similarly flexible approach are able to study simultaneously in all four traditions and to follow the practices of all four. The Dalai Lama is an example. So also is Ringu Tulku. And here he talks about how you can use this approach with all religions and paths, About how it is respecting the diversity, and that differences are good.

For the historical background to this approach in Tibetan Buddhism,s see Ringu Tulku's introduction "What is Rime" in his book "The Rime Philosophy of Jamgong Kongtrul the Greats.

Ringu Tulku is a teacher who doesn't put on airs and is very modest and humble but also counts amongst the more learned of the Tibetan teachers, a professor of Tibetology who has also mastered the teachings and practices of all the four schools of Tibetan Buddhism.

To actually practice two different paths wholeheartedly at once is only for people with very flexible minds, and there are many impediments in the way for most of us. However much one might want to do it in some cases, most of us just can't. I know I couldn't do it, even though I was brought up a Christian and practiced wholeheartedly as a Christian and now practice whole wholeheartedly as a Buddhist. It took me several years to make the transition and the idea of practicing both at once is way beyond me.

And - this is not a Borg type approach of assimilation - you know, how the Borg in Star Trek say

"We are the Borg. Lower your shields and surrender your ships. We will add your biological and technological distinctiveness to our own. Your culture will adapt to service us. Resistance is futile."

That’s not respecting and valuing the differences. That’s like the Vikings looting the sacred books of Christian monasteries for their gold and jewels. You might end up with something of value for your religion, but without its original context and inspiration, often it ends up as a hodge podge of ideas grabbed from here, there and everywhere.

But we can all respect other paths, and approaches, and the integrity of the people who follow them, and value the differences, and enjoy talking to others of other religions and be enriched by the experience, learning about how they understand things and their path and talking about our own path with them. Also, I think it’s inspiring to all of us that it’s possible at all to practice two such distinctive religions at the same time.

Neil Hiatt in his answer to Can I be both Christian and Buddhist? says this is not possible because the Bible says you can’t be both. For some people, it is impossible, unthinkable even. That's part of their path. So you need to respect that. To do otherwise is not respecting and valuing the differences.

However it depends on how you think about the Bible, and also the Bible often contradicts itself. It's not a legal document, it's a collection of writings that Christians think is inspired by God.

If you say it is something Christians can’t do, do you mean "According to God", or "According to Jesus", if so how do you know? Do you mean "According to the Bible"? If so it is interpreted in many different ways. Do you mean "According to some particular Christian teacher who you respect, minister, pastor or some such"? If so, well it's your choice to follow their path but other Christians don't have to. Same also if you just mean "according to my own interpretation of Christianity".

And it’s the same the other way too. Buddhists do have beliefs and faith and commitments. The main one is that Buddha outlined a path that can lead one to a truth that one can come to see for oneself. It's a commitment to a particular path and approach. If you look at it on paper like a legal document, then it's inconsistent with Christianity too, to commit to revealed truth in a scripture, to say that you know there is a God, to say that you know that Jesus loves you, to say that you will go to heaven when you die, that you know that because the Bible says so, anything like that is saying that you have decided not to have an open mind in particular topic areas - at least, if you say you believe that in the sense that you know that there is no need to look at it any more because it is decided for you already. That's going against the openness of the Buddhist path.

So there are issues that way too. That's the main reason I'd find it really hard to practice the Christian path at the same time as the Buddhist path, because how can you do that and a the same time keep the openness of the Buddhist path? It's like as a Christian you can question almost anything depending what path you follow but there is always some core of things you are sure of, even if you doubt it, your aim is to see through those doubts and to get back to faith in those things.

But if you have the flexibility of mind and approach, you can handle that and it's not a problem for you. Somehow.

And after all you can be brought up as one and then transition to the other. What's to stop you from doing that transition quickly, even many times a day? Just that ones mind isn't flexible enough. If your mind has the flexibility, what's to stop you from following both paths at once - doing Christian practices wholeheartedly as a Christian and Buddhist practices wholeheartedly as a Buddhist? And if you find that you can integrate them both in your life, contradictory though that seems to others - well if you find you can do it and it causes no problems and you find they help and enrich each other in your life, again what’s the problem? It could be like friends from different religions working together in their lives towards something they couldn’t achieve separately.

Unless someone comes along from your church or temple and says "Look you have to stop doing this or you'll be expelled and we won't let you in any more" in which case it is a society thing rather than anything intrinsic to the religious paths you are following.

Elaine MacInnes shows by example that it is possible.