The existence of a wandering preacher that taught essentially the ideas of the Pharisees and was put to death by Roman authorities afraid of an insurrection doesn't seem far-fetched. In fact, we know of several other people exactly like that. The difficulty is separating the historical person from the myths that quickly surrounded him. His original followers were primarily Jewish, but after Paul got into the act, things changed drastically. The Roman take-over of Jerusalem also affected the politics between the groups.