The Bible doesn't say anything about quantum physics, calculus or innumerable other subjects either.
Near-Death Experiences and reincarnation are two entirely different subjects. By definition, an NDE is not a "death" experience. There is nothing inherently unbiblical about NDEs. The basic phenomenon as originally described by Raymond Moody in Life After Life is remarkably consistent with Christian theology. One of the most noted NDE researchers, Dr. Michael Sabom, is a fundamentalist Christian. One of the foremost experts on the Resurrection, Gary Habermas, is extremely interested in and excited about the NDE phenomenon.
The content of many NDEs is not consistent with Christian theology. The theology that some NDE groups develop on the basis of the content of such NDEs is not consistent with Christian theology, being more Buddhist or New Age in nature. However, many NDE experiencers do integrate their experiences into their Christian beliefs. Others are led to Christianity through their NDEs. Others abandon Christianity as the result of their NDEs. I know one woman who experienced two profound NDEs and is a militant atheist. NDEs are simply a phenomenon, a body of evidence to be dealt with like any other body of evidence.
Reincarnation is deemed by many to be inconsistent with Christian theology, largely on the basis of Hebrews 9:27. When Hebrews 9:27 is read in context, it seems to me a weak basis on which to disregard a pretty substantial mountain of evidence pointing toward reincarnation, but each person must do what he or she will with the evidence. Certainly a number of doctrinal difficulties would be eliminated if reincarnation were true.
Regardless of the extent to which the NDE phenomenon and reincarnation do or do not have support in the Bible, they are both part of the reality in which we live. The NDE phenomenon, as a phenomenon, is undeniable - it exists, it is evidence of something, and it is studied by researchers and scholars of the highest caliber. To some extent the same is true of reincarnation - researchers and medical professionals of the highest caliber, such as the late Dr. Ian Stevenson of the University of Virginia, have amassed a huge body of anecdotal evidence (and even medical evidence, such as Dr. Stevenson's work with birthmarks).
One reason many Christians exist in a state of cognitive dissonance is this bizarre attitude that any aspect of reality which doesn't correspond to something in the Bible must be suspect or even demonic. At the extremes, they feel they must believe (or at least pretend to believe) the earth is flat and/or 6,000 years old because "this is what the Bible says" when the rest of the world (including most of Christendom) knows these things are not simply false but laughably delusional. At a lesser extreme, they hide from evidence, such as that for NDEs or reincarnation, that they fear might challenge their "biblical" worldview.