It is not that I regard reincarnation as impossible, rather that I believe it is probably very rare - and the reason I believe this is related to incarnation being irreversible.
We start-out as pre-mortal spirits - and we incarnate in order to make progress towards full divinity.
All Christians at least implicitly believe that to be resurrected - that is, to be incarnate, to die and then to be incarnated again with a perfected body - is 'better' in some vital way than simply to be 'a spirit'. That to be resurrected is a higher state than to be a spirit - otherwise why go to the bother of incarnation and death.
It also seems that there is a very general folk wisdom, spread across many religions and spiritual practises - that to die is to separate spirit from body, but that to be a spirit whose body has died is to be in some way maimed, incomplete, miserable, and indeed to be unselfed. This leads to the 'underworld' of post-mortem spirits - Hades, Sheol and the like - a world of partial and demented spirits, living in an eternal and unpleasant present.
What I take from this is that incarnation is progression, and it is also irreversible - once a spirit has had a body, the body cannot afterwards be detached from that spirit without some maiming, some irreparable damage.
Now - what this seems to mean for reincarnation is that it has to involve 'the same' person coming back. I think this is entailed, because the body would (I think) have to be remade from the surviving spirit - in something I imagine to resemble a complementary process.
In other words after death there is a maimed and incomplete spirit, and resurrection entails re-completing it with 'the same' body it had during life, but this time an immortal, perfect and pure body.
If this person was reincarnated then either they would have to return to earth with this immortal body - in which case they would be an incarnate angel rather than a resurrected human. An example would be the Moroni; who is an important human character featured in The Book of Mormon, and who then becomes the angelic agent for the rediscovery and traslation of the book by Joseph Smith.
(Note: There are also thought to be angels who are pre-incarnate or never-incarnated spirits.)
A reincarnated human would, I take it, have to be re-born to human parents - and if a post-mortem spirit was indeed reborn in this way he would need to be provided with a new body that was nonetheless in some essential way the same body he had before - not necessarily the same in appearance, but the same in some essential fashion; because otherwise he would remain maimed; and also otherwise because if he had a different body when reincarnated, then he would not be the same person somehow reborn, but someone fundamentally different.
So I can imagine that a reincarnate might arise when (for whatever reason, perhaps a premature death such as being murdered - premature in terms of what they had been incarnated to accomplish, in a spiritual sense) - would instead of being resurrected, have their spirit 're-cycled' t be born again - but this recycling would be the same person, with a body that was the same in its ultimate essential quality (even if it did not look identical).
I expect that this thing has happened, and continues to happen - but such an idea of reincarnation apparently rules out some of the attributes and things it is supposed to achieve in Eastern religions. It seems to rule out incarnation as other (non human) beings, and also the idea of reincarnation as a way of gathering very different experiences of being different kinds of person.
I think reincarnation is more of a second chance (or maybe third, fourth etc chance) to do what needs to be done - rather than a mechanism for incremental, stepwise spiritual progress. And this conviction of mine comes from my understanding of what happens to the spirit at death and resurrection.