Well, according to people who have done past-life regression hypnosis, and the people who have recorded and shared about it, people seem to be reincarnated in a variety in times and places over the centuries, but no one AFAIK has reported regressing to a past life that was at a future date. That’s one of several major reasons that would have me say no.
(P.S.: I just now read @ragingloli‘s second answer above, which says this same point.)
I’d make a half-joke about how humans are about to wipe themselves out, so that might explain it, but regressions have been reported for decades, and the spread of reported times suggests to me that there would be some people reporting future regressions even if we’re about to go extinct.
Another major reason I’d say no is that I think backwards time travel in the same slice of the multiverse is paradoxical as it would or could contradict the timeline that led to that travel, even, I think, if it were only via reincarnation.
However there are two senses in which I would not say no:
One is if the multiverse exists as an infinite continuum of possibilities. In that case, I suppose it could theoretically be possible to reincarnate into another similar but earlier part of the multiverse, maybe. No one really knows if or how that (the multiverse part) might be the case or work, but serious cosmological theorists do mention such ideas. And I don’t see the same paradoxes with that that I mentioned above.
Another, is the perspective available from certain philosophies and theories that the ultimate reality is not just that we are falling steadily through time. But that consciousness shifts attention from time and place to time and place. In that sense, I think consciousness does shift attention and may be able to go back to past times. But that’s not about reincarnation per se, and so while a consciousness now might shift its attention to an earlier time, the perspective at that time would have no access to the memories from the later time.