Reincarnation is a very complex subject. People who say it's true and people who say it's not true can have identical beliefs. I'm not sure our time on earth provide us with the experiences that are needed to correctly understand it.
Silver Birch had this to say:
"There are also what I call facets of the one diamond. This is the over-soul, the greater individuality, and the facets are aspects of it which incarnate into your world for experiences that will add lustre to the diamond when they return to it.
Also there are people who, although separate persons, are aspects of the one individuality. For instance, my medium, his wife and myself are parts of one individual. So you can have facets of the one guide. You can call these extensions if you like, but it comes to the same thing. Only an infinitesimal part of the whole individuality can be manifested in physical form on earth."
I have seen this notion of spirits retaining individuality but forming higher level entity in other writings too and I don't really understand it. However, there is reason to say that reincarnation is true and also reason to say that it is not true. It might be that the individuals don't reincarnate but the "over-soul" does send some part of itself to incarnate again. It begs the question, "what am I?" After death, is there a time when the individual merges back into the oversoul and is that like regaining lost memories? I don't know. I wonder if consciousnesses can merge to form a new entity while the individuals that make it up retain their individuality. I don't know if that is true. One explanation of God is that at the highest level of the spirit world advanced beings form a collective consciousness while retaining their individualities.
When you ask if reincarnation is true you also need to understand that there are different doctrines of reincarnation. Theosophists believe that the spirit goes into a sort of nether region until they reincarnate rather than returning to a spirit world and studying their and preparing for subsequent life. Having that wrong belief about reincarnation in life leads spirits to remain earthbound and obsess children in their attempts to reincarnate. (See Thirty Years Among The Dead by Carl Wickland http://www.spiritwritings.com/ThirtyYearsAmongTheDead.html)
When people who come from this philosophical background ask spirits if reincarnation is true, the spirits may very well tell them that reincarnation is not true because of the above ideas expressed by silver Birch and because the wrong view of reincarnation has dangerous consequences.