Continuity mistake: In the scene where the postman gallops by the country house full blast and then comes back to pick up the letter from the boy's hand, you hear hoof sounds for long enough to take him hundreds of metres away, while the camera is on the boy. But when he is actually shown stopping to turn his horse around he has only come as far as the corner of the premises, maybe 20 meters away.
Continuity mistake: For most of the time Kevin Costner has a beard, it is scraggly and grows unevenly on the sides. In one scene where he appears as the Postman early on, it is perfectly combed and there is hair where he doesn't have any naturally. I would guess they had to reshoot the scene and he was already clean shaven, so they had to "create" a beard to maintain continuity.