Let us assume that humans posses the ability to percieve beyond the bounds of the 5 accepted senses. Be it a gut feeling, Nostradomus, the psychic friends network, or deja vu we've all had an experience with an extra-sensory phenomenon.
Let us also assume that this ability is determined by our genes. There is evidence pointing towards this. A family in California is reputed to produce generation after generation of psychic kids.
Let us also assume that the more similar your genes are to another persons the stronger the "psychic link" with this person. Twin studies suggest a level of similarity shared between twins (separated at birth and raised in radically different environments) too high to be attributed merely to coincidence.
Let us note that Genetic Reincarnation is proven mathematically and its occurance in humanity is only a matter of time and population. Genetic Saturation, wherein all codes are currently in use (possibly multiple times), occurs in mitochondria... though the implications of a reincarnatied mitochondrium are unknown at this point.
Let us also note that all of the above assumptions can be subjected to the scientific method and falsified if, indeed, they're false.
Prophets: They "see" through the eyes of a future incarnation
Exceptionally Good Historians: They "see" through the eyes of a past incarnaion
The Immortal Soul: You may not have a soul of your very own, but you do have a genetic code unique unto yourself.
Karma (in the New Age sense): You do something evil, mean, bad, and nasty to someone in this incarnation. When you meet in a different incarnation they "remember" and are more disposed to be evil, mean, bad, and nasty towards you in return.
Telepathy: This is actually one I came up with whilst arguing this with a relative. She has a strange affinity for the term "vibrational frequency." (IE, the most advanced people, psychically speaking, have the highest "vibrational frequency") It made me consider the quantum vibration and that, given the different masses of the neucleotide bases they would, in turn, have different quantum vibrations. She also has an affinity for the term "on the same wavelength." People with genetic codes that were similar would, quite literally, have DNA molecules "on the same wavelength" with similar quantum vibrations. If similarities between genetic codes is directly related to the strength of a "psychic link"(Assumption 3) a lesser level of unspoken communication may take place between dissimilar people.
The Advance of Humanity: Knowledge is Cummulative, obviously. However, when a new idea is introduced it is often questioned against the old ideas, feared, banished, etc. But ACCEPTANCE may also be cummulative. Down the road of incarnations the once new idea becomes an old idea, and the second time around one might be more prepared to listen with reason.
Schitzophrenia (as differntiated from "dissociative disorder"): someone with an unnatural reception to the experiences of other incarnations would be subjected to the constant bombardment of thoughts and conversations not literally their own.
A child's "imaginary friend" could be far less imaginary than you might think.