There were many claimants to be Messiah, actually. The key issue is that the classical reading of messianic prophecy was that the coming Messiah would be first and foremost a political-military leader who would throw off foreign domination, establish an independent Jewish state under their rule, and secure the traditional practices of the Jewish religion. The vast majority of claimants to be Messiah attempted to follow this model.
The problem for such claimants, however, is that the period of greatest messianic expectation occurred during Roman domination, and so such men were usually dispatched by a far superior military power. Since the model of Messiah they proposed to fulfill depended on political-military victory, their deaths generally meant the end of their movement.
Christian Scriptures actually mention a few such figures. Acts of the Apostles, a history of the earliest Christian community, describes Jewish leaders discussing how to deal with the burgeoning following of Jesus. One among their number, Gamaliel, argues for tolerance on the assumption that such a movement should dissipate on its own if it is indeed false. To support this reasoning, he cites two recent messianic claimants: Theudas and Judas of Galilee, both of whom gathered followings, were killed, and failed to have a lasting impact.
The most famous and successful example of a messianic movement besides that of Jesus of Nazareth would have to be that of Simon bar Kochba, who led a revolt against Roman authority in the 130s AD. Bar Kochba does not appear to have initially identified himself as Messiah, but a messianic movement arose around him, promoted by the well-regarded Rabbi Akiva, who was also instrumental in helping define the nature of post-Temple Judaism and expected Simon bar Kochba to restore the Temple should the revolt succeed. However, bar Kochba's Jewish state only lasted a few years before brutal Roman suppression destroyed it and killed its leadership. In the scourging of Judea and the Jewish people that followed, most renounced Simon bar Kochba as the Messiah they were expecting.
There is a messianic movement that has persisted to the present day, however. Mandaeism, a gnostic and dualist religion mostly practiced in Iraq, recognizes John the Baptist (Yahia Yohanna) as the true Messiah and the greatest of the prophets. However, they number less than one hundred thousand and I don't know enough about their movement to comment further on their concept of Messiah or other doctrines.