Actually, it's the Christian version of the Nativity that is off. Jesus wasn't from Bethlehem, he was a Nazarene (Jesus of Nazareth). The Nativity story also only appears in the books of Matthew and Luke, and there are considerable contradictions between the two accounts.

For one, Luke contends that Mary and Joseph traveled to Bethlehem (the town of his ancestors) to take part in the census of Quirinius in 6 CE. However, that makes no sense. The Roman did not conduct censuses based on where people were born, but where they lived!

Matthew claims there were three wisemen who came to Jerusalem looking for the "king of the Jews," which prompted Herod to murder the firstborn in Bethelehem. None of this is contained in Luke's account. Also Herod died between 4 and 1 BCE - 7 to 11 years before the census.

Matthew claims that the family then fled to Egypt to avoid being murdered. Luke claims they travelled to Jerusalem (the heart of Herod's kindgom) to get a blessing. Which was it?

The obvious explanation is that over the course of several generations before the Acts of the Apostles were written, his followers created the Nativity story to make it consistent with prophecy (the Book of Micah), much like the claims that he carried out miracles.