As much as one can point to passages like Hebrews 9:27, people can interpret that in various ways, or patently ignore it.
If you have to believe everything and follow everything from the bible consistently, then there are no such things as christians. Can you be a christian and have wealth? Can you be a christian and not cut your hair a certain way? Can you be a christian and be lustful?
No. To say that Reincarnation is true is to say that the teachings in the New Testament are wrong (Hebrews 9:27), which invalidates you as a Christian. In fact, this is one of the major fundamental differences between Christianity and Mormonism. Mormonism is a supporter of the view that reincarnation occurs after death, and so goes a perpetual cycle of rebirths until you are perfect enough to be a god of a universe.