For some perspective, just consider the response of Ramana Maharishi, who was in the view of many the greatest of the Hindu sages to have lived in modern times (died 1960):
Is reincarnation true?
Sri Ramana Maharshi:
Reincarnation exists only so long as there is ignorance. There is really no reincarnation at all, either now or before. Nor will there be any hereafter. This is the truth.
[Note: Comments by David Godman: Most religions have constructed elaborate theories which purport to explain what happens to the individual soul after the death of the body. Some claim that the soul goes to heaven or hell while others claim that it is reincarnated in a new body.
Sri Ramana Maharshi taught that all such theories are based on the false assumption that the individual self
or soul is real; once this illusion is seen through, the whole superstructure of after-life theories collapses. From the standpoint of the Self, there is no birth or death, no heaven or hell, and no reincarnation.
As a concession to those who were unable to assimilate the implications of this truth, Sri Ramana would sometimes admit that reincarnation existed. In replying to such people he would say that if one imagined that the individual self was real, then that imaginary self would persist after death and that eventually it would identify with a new body and a new life. The whole process, he said, is sustained by the tendency of the mind to identify itself with a body. Once the limiting illusion of mind is transcended, identification with the body ceases, and all theories about death and reincarnation are found to be inapplicable.]
Can a yogi know his past lives?
Do you know the present life that you wish to know the past? Find the present, then the rest will follow. Even with our present limited knowledge, you suffer so much. Why should you burden yourself with more knowledge? Is it to suffer more?
When seen through the sight of the supreme space of Self, the illusion of taking birth in this mirage-like false world is found to be nothing but the egotistical ignorance of identifying a body as "I". Among those whose minds are possessed with forgetfulness of Self, those who are born will die and those who die will be born again. But know that those whose minds are dead, having known the glorious Supreme Reality, will remain only there in that elevated state of reality, devoid of both birth and death. Forgetting Self, mistaking the body for Self, taking innumerable births, and at last knowing Self and being Self is just like waking from a dream of wandering all over the world.