Technically we can't know either way. However it's a giant leap from consciousness stored in a finite material brain to consciousness that could exist separate from said brain let alone somehow move from body to body. That seems like an awful lot of wishful thinking by people who can't accept their own mortality so they make up stories that give them a sort of emotional relief.
Clearly our consciousness as human-beings is intimately tied to our brain: if parts of it are destroyed or damaged our consciousness is diminished or altered. Since this can hardly be disputed how did it get there assuming reincarnation is real? Why would it take maturation of the brain to fully express this consciousness to the point of abstract reasoning, complex emotions and the deep seated belief in individuality or even reincarnation?
You'd think that if reincarnation is real consciousness must be immaterial: how can the non-material (consciousness) interact with the material (the body)? The French philosopher Descartes assumed duality according to his christian beliefs: humans are made up of a body and a soul. Yet he completely failed to explain how this interaction was even possible and his hypothesis (soul and body meet in the pineal gland in the brain) is pretty much ludicrous. He probably only came up with it because there was no explanation for why it was even there in the first place.
Believing in something doesn't make it true. Since there is no evidence reincarnation is even remotely possible I simply assume death is the final and permanent end to individuals. Occam's razor (entities should not be multiplied without necessity meaning the simplest explanation is theoretically preferable) would suggest a negative answer to the question.
In any case we don't know how consciousness arose but that does not make reincarnation any more likely since reincarnation can't be explained either. How can we possibly know we were ever alive before being born?
This isn't related to your question but I don't get the appeal of reincarnation: who would want to be reborn in any case? Isn't one life enough to convince anyone of its perfect meaninglessness and uselessness? Even the religions that are built upon the assumption of reincarnation claim it's better to escape life altogether although their explanations about how this could possibly be achieved don't exactly make sense.
