Are you talking about the “Golden Urn” ceremony? If so, it was only used to select the eleventh Dalai Lama according to the current Dalai Lama.
“The Golden Urn system was actually used only in the cases of the Eleventh and Twelfth Dalai Lamas. However, the Twelfth Dalai Lama had already been recognized before the procedure was employed. Therefore, there has only been one occasion when a Dalai Lama was recognized by using this method. Likewise, among the reincarnations of the Panchen Lama, apart from the Eighth and the Ninth, there have been no instances of this method being employed. This system was imposed by the Manchus, but Tibetans had no faith in it because it lacked any spiritual quality. However, if it were to be used honestly, it seems that we could consider it as similar to the manner of divination employing the dough-ball method (zen tak).”
The Office of His Holiness The Dalai Lama
So, such a ceremony could be used as a form of divination, but the Tibetans had no faith in it and it was only used once in this way, other times they pretended to have used it, but had selected the Dalai Lama beforehand.
More recently the Chinese used it for the Panchen Lama (not the Dalai Lama), but the Dalai Lama had already chosen a different child as the next Panchen Lama and the Dalai Lama’s choice disappeared 21 years ago, aged six and has never been seen since.China urged to release Panchen Lama after 20 years - BBC News
So anyway in the past, yes seems they generally have been successful unless you count the eleventh Dalai Lama but that could have been a successful divination. But with the Panchen Lama, then no, the system has lead to the most recent of the Tibetan choices for the Panchen Lama disappearing at the age of six.
The current Dalai Lama has said he won’t take rebirth in China, I think understandably so in the circumstances, and more recently, has said that he may also be the last of the Dalai Lamas. He says it depends on the situation when he dies and the will of the Tibetan people.
This has lead to the rather strange situation of the Chinese government telling the Dalai Lama that he has to take rebirth, and that his rebirth has to be in China and abide by Chinese regulations about rebirths, and the Karmapa saying “In my view, it is only the Dalai Lama himself who should decide about his future reincarnation. So I am confident and have full trust in his decision. “
It's up to Dalai Lama whether he'll be reborn, Buddhist leader says
It’s amusing, in a sad kind of a way, since the members of the Chinese government, as far as I know, usually don’t believe in rebirth. (The emperors of China before this present government also didn’t believe in rebirth as they were Confucian).
Whether or not he takes rebirth as the next Dalai Lama, there’s nothing to stop him taking rebirth in some other form. The dispute is about whether he will take rebirth as a reognized Dalai Lama and continue the institution of the Dalai Lama.
He jokes that it might be a good time to end the institution of the Dalai Lama with a relatively popular Dalai Lama and the next one might be stupid or do something that disgraces the institution. The Buddhist ideas of rebirth don’t mean at all that successive rebirths are the “same personality”. They often have very different inclinations and personalities.For instance the sixth Dalai Lama was a poet, writing inspiring poems that are much loved by Tibetans to this day, never became a fully ordained monk and instead gave up the monk’s robes, and had many girlfriends in his short life. While the present Dalai Lama has a strong inclination to study, passed his exams in Buddhist scholarship at a young age with flying colours, impressing all the teachers who were judging him, and is of course a celebate monk and is also very interested in Western science, amongst other things..
More about it here, quote from Chonpel Tsering, the Dalai Lama's representative in northern Europe.:
"I genuinely don't know [whether he will reincarnate]. Tibetan Buddhism believes that everybody is reborn, but not everybody can choose how and when they are reborn.
"The lamas - the senior religious figures - are able to determine firstly whether they are reborn, and if they are going to be reborn, where they'll be reborn.
"The present carnation, the present Dalai Lama, can decide. The rebirth is his choice.
"First, he is going to consult the Tibetan people and others that follow Tibetan Buddhism to find out whether they think that there should be a 15th Dalai Lama.
"If the decision is, 'Yes,' then he will set out clear instructions about the process, so that there is no ambiguity, so that the reincarnation process isn't manipulated or misused by anybody for their own personal or political interests.
"The Chinese and the Communist Party have set up systems where reincarnations such as his holiness the Dalai Lama have to be recognised and approved by the Communist Party.
"Maybe [they] feel that if they pick their own 15th Dalai Lama, somehow that authority will transfer on to the [one] lama they have picked.
"His holiness has said that the 15th would be born outside of Tibet, outside of China, because this 15th Dalai Lama would have to continue the work of the present Dalai Lama.
"If there is no genuine religious freedom inside Tibet, then it's very hard for him to continue that work."
HOW DOES THE CHOICE TO TAKE REBIRTH TAKE, AS UNDERSTOOD BY TIBETAN BUDDHISTS?
This is just a crude explanation, which hopefully will help readers who haven’t come across this before.
The idea is that some of the Lamas such as the Dalai Lama can direct their next rebirth. As they die, then most of us, so the Tibetan Buddhists believe, enter into a Bardo state.
(Therevadhan Buddhists have a different idea of how rebirth happens - they think the end of one life is the first moment of the next. Tibetan Buddhists think this is true in some rare cases but that most of us enter the Bardo for a while before we take rebirth, and that sometimes you can be in the Bardo for up to several weeks).
Once in the Bardo, you have lost your previous bodily form and experience extremely bright lights,loud sounds, and most get terrified at that point and rush away and try to hide in a corner, and so take rebirth again. But the idea is that there are some who are able to relate to what’s happening, not necessarily Buddhas, who can direct their next rebirth. They then take rebirth where they feel they are needed or can continue their work.
So that’s where that idea comes from that the Dalai Lama because he feels he wouldn’t be able to continue his work within China because of the restrictions on religious freedom, would choose to take rebirth somewhere outside of China.
It’s not the young child who makes that choice. It’s the previous Dalai Lama as he dies. Before he dies he often will leave some instructions about how to find his next rebirth. Also sometimes there are indications in dreams to help them find him. Then the new child is tested in various ways to make sure he really is the next Dalai Lama, as a check against mistakes in the process.
That’s why it’s quite strange for someone who believes in rebirth, that the Chinese government would feel they can regulate this. How can they legislate for the decisions someone makes as they die in the Bardo state? They can’t even choose whether the Dalai Lama turns left or right as he leaves the entrance to whatever place he stays for the night. So how could they decide what he chooses to do in the Bardo state?
(Added this in response to a question in the comments)
NO POLITICAL ROLE FOR THE DALAI LAMA
The Dalai Lama divested himself of all political roles in the “government in exile” in 2012.
“He also changed the constitution and made the final implementation of a change that no lama will be head of state in any future government that Tibetans approve of.”
The Dalai Lama Will Not Return to Lead Tibet (He Has Something Better in Mind)
So there is no need at all for the Chinese to be concerned about the Dalai Lama or any of his future rebirths becoming a head of state of the government in exile as Dalai Lama. Not only that, no other reincarnate Lama can either.