There are a few reasons I suspect reincarnation is both possible and exists:
1. As a child I had past life memories (I don't remember anything now). I stopped talking about it when no one believed me.
2. It is the best explanation for other children who have veridical past life memories.
3. It explains certain things about human development that we don't understand.
4. It makes sense mechanically, working as a sort of metaphysical version of gravity.
5. It's fair.
6. Pretty much everyone believed in reincarnation in some form until the 14th century. (Reincarnation, of at least certain individuals who suffer traumatic deaths, is still a part of orthodox Judaism today.)
The main questions that exist surrounding the idea of reincarnation are as follows:
1. Which of the two suggested mechanisms for reincarnation is accurate, the individual soul coming back life after life as all the world's religions suggest, or the new age model where different souls incarnate that are connected through an "over-soul"?
2. How long is the typical duration between successive incarnations on Earth?
There seems to be a great deal of leeway with this, depending on the belief of the person. Some people who believe in instantaneous reincarnation may indeed instantaneously reincarnate. In the Tibetan model rebirth takes place after 45 days, ideally, though with reincarnated lamas it looks like it usually takes 2 or 3 years. With a lot of people from the Second World War and the Holocaust they seem to have started coming back after 50 or 60 years. Yogananda suggests 200 years is a normal time. I suspect that people wait to reincarnate until after the whole family has died, so they can all meet up again before departing once more for Earth, although with sudden horrific deaths the wait time seems to be a lot shorter.
An enlightened being can reincarnate at will, but with the average person I suspect something akin to the Hindu system is in place where a person spends time in one of the astral heavens until the merit they accrued in the previous lifetime has expired (which may take hundreds or even thousands of years) before having to come back. This makes sense, metaphysically.
3. How many souls are there that are incarnating? Let's for the moment just talk about humans. (I suspect more advanced animals like apes and dolphins also reincarnate like humans, but we can't ask them so it's impossible to know. Also animals who have spent a great deal of time with humans, like pets, also may have individual rebirths. Generally it is said that wild animals, and I suspect plants too since recent studies with tomatoes and trees reveal that at least certain plants are conscious too, tend not to have individual souls, but I have no idea.) There are at least 8 billion human souls, and since the maximum number of humans possible on the Earth is assumed to be about 12 billion let's say that's the minimum number of individual souls. That would mean that the time between incarnations would necessarily be very short. If, on the other hand, there were 100 billion souls then the time between incarnations would be very long, since there have only been 108 billion people who have ever lived, so only a handful would have ever reincarnated. If there are 1 trillion souls then no one will ever need to reincarnate because it will take millions of years to run out and hopefully humanity will all reach enlightenment before then and we can be done with the Earth permanently.
4. Why are all, or most, of the faculties lost in the interim between incarnations? Babies, even of enlightened beings, still have to learn to walk and talk all over again, and very few people come here knowing calculus (reincarnation is, I think, the only explanation for genius as seen in people like Freeman Dyson who knew more about mathematics as a very young child than most people ever learn in a lifetime). You would think someone like the Dalai Lama who has come back 14 times now would be born knowing how to walk and talk, because he has that much more experience than an ordinary mortal.
It does seem that some memories sneak through, at least at the beginning. Young children exhibit xenoglossy, speaking languages they never learned, but they knew from a previous life. This universally disappears after 4 or 5 years. And with genius and rapid development of certain faculties there seems to be some degree of anemnesis at play. If you learned something ten times already then relearning it in your next life will be very easy.
These lists are in no way exhaustive. Rather, they serve as a useful summary of the situation.