When Jesus asked Peter who people said he was, he answered that some people thought that Jesus was Elijah come back. Peter knew better and said he was the Christ. In any event, John the Baptist himself directly denied the claim (see John 1:19-21).
What Jesus more likely was saying in Matthew 11 is that John the Baptist was a prophet in the tradition of Elijah. Elijah was the greatest of the prophets (even if his disciple Elisha inherited a double portion of his spirit). All of the prophets that followed came in the name and spirit of Elijah, in the same way that some of Paul's letters apparently were created after his death. They were in the spirit and tradition even if not the actual man.
Note also Elijah never died - he was taken up into heaven, and people thought he would return. As such, no Elijah was not being thought to have reincarnated- only returned.
Elijah's return was long a part of Jewish custom. Even today, during a Passover sedar, a chair is left empty for Elijah. But again, since he did not die, Jews look for his return, not his reincarnation.