Anthony asked what logical possibility is. I'm not sure, but I think a proposition is logically possible if it does not involve or imply a contradiction. In that sense, is the proposition "I have been reincarnated" logically possible? Does it fail to involve or imply any contradiction?
I think it is logically possible. The key assumption is that the term 'I' does not refer to my body alone. For being reincarnated means having once had a body that is numerically different from the one I have now. I say 'numerically different' because obviously my body was qualitatively different from how it is now. It used to weigh somewhat less, for example. So, having a body that was numerically different from the one I have now, means not identical to the body I had in 1980, or 1970. I don't think there is anything in the reference or meaning of 'I' that entails such an identity.
That's not to countenance disembodied egos or anything like that. The possibility of reincarnation does not require there to be a disembodied referent for 'I'. But if there are no disembodied egos, and if reincarnation takes place some time after the death of the previous body, there has to be a time when the 'I' does not exist. E.g. suppose my body used to be Napoleon's body. He died in (er, looks in Wikipedia) 1821. I was born in 19xx. So if that were the case, my ego would have temporarily ceased to exist in 1821, then was recreated in 19xx.
Does that mean I am the same person as Napoleon, if that were true? On the assumption that two egos cannot own the same body, then yes. Does reincarnation violate any basic non-logical principles, such as the principle of sufficient reason, or Ockham's principle? More later.