For follow-up discussion and general commentary on the topic. Comments are sorted chronologically.
I believe that there is no reincarnation and also I believe that the Bible does not support reincarnation.
But there are instances when some people claim that they or someone remembered things and relationships of past life. Such instances, I believe, are satanic deceptions. That is, a demonic spirit is playing mischief by bringing the memories of any past life it had possessed previously to someone whom it is presently possessing. The incident of Acts 19:14-16 may be cited to support this theory. The person here possessed by the evil spirit might not have known Jesus or Paul personally but the evil spirit knew both of them.
Could this be a possible explanation?
My belief it is demons that want you to believe you had a past life. You are experiencing their previous lives. When you get saved those memories leave for the most part, if not then you need to be delivered. Many of us christians still have spirits in us because we have not been delivered. I had some come out after I had a deliverance.
It is important to understand that such "memories" are not in the conscious mind, they are "drawn out" while the person is in a hypnotic state. To be hypnotised is to be in a very relaxed and compliant state, in a highly suggestible frame of mind.
Studies have been done that used the same hypnotic techniques to not only "reveal" memories of past lives but also of FUTURE lives!
There is also the fact that so many "remember" being the same past person, nearly always a famous historical figure. Poor Cleopatra and Napoleon must have had a terrible time being the reincarnation of hundreds of people at once.
Also, logically, given the current world population, there are too many alive now for all to be previously reincarnated, if our "current" life began as soon as the previous one ended.