Key Verses: Hebrews 9:27 And as it is appointed unto men once to die, but after this the judgment:
John 11:43 And when he thus had spoken, he cried with a loud voice, Lazarus, come forth. 44 And he that was dead came forth, bound hand and foot with graveclothes: and his face was bound about with a napkin. Jesus saith unto them, Loose him, and let him go.
Matthew 27:52 And the graves were opened; and many bodies of the
saints which slept arose,
53 And came out of the graves after his resurrection, and went into the holy
city, and appeared unto many.
The one word answer to this question is "No". In spite of many Hollywood movies who picture the dead as coming back for a second chance, and Eastern religions that teach an endless lives leading to some hoped-for oblivion, the Bible is quite clear: This life is our only chance to get it right; if we don't accept Christ in the bodies we have now, we won't get another one later on. The doctrine of reincarnation is, at its heart, an indulgence in wishful thinking, and an excuse for license, just like the doctrine of universal salvation, which is also not scriptural.
This does not mean we will never know physical life again. The Bible has several instances of people's being raised from the dead, but, with the exception of Christ, they all died again. All of us, though, will one day be raised up in new bodies, whether to eternal life or eternal death. In a very broad sense, that does meet the definition of reincarnation, but not in this present world. The bodies we will have, will not be some ethereal spirit bodies, but flesh and blood. We will be like Jesus, whose resurrected body could be touched (John 20:27), and who ate food with his disciples (John 21:12-15).
Not only can people not be reincarnated as other people, they can't be reincarnated as animals, or vice versa. Only man was created with a living soul (Genesis 2:7), and without the soul, there is no human life. Animals don't go to heaven, and animals do not hide within them the spark of existence that God gave to man. There is no such thing as a sacred animal, except an animal offered to God under the old sacrificial system.
Reincarnation is a distraction. Any belief or doctrine that directs people's attention away from the only real hope of new life, Jesus Christ, is of the Devil, what Paul calls "doctrines of devils" and "seducing spirits" (1 Timothy 4:1). There are a multitude of false doctrines in the world, but only one true Gospel. The sad thing is that every new, or old, false religion or doctrine has its adherents, one more way to keep people away from a saving knowledge of Christ. This is consistent with Christ's teaching that only a few will enter into the narrow gate that leads to the Father (Matthew 7:13-14).