Revealing mistake: When the elephants are making fun of Dumbo in the train caboose, Dumbo's mother pulls a pin which causes the stall door above the other elephants to fall. When the door bounces back up, you can see that the animators forgot to draw the elephants' heads and trunks and only their bodies are visible.
Continuity mistake: At the beginning, Belle dismounts from the horse and gets in front of Mammy Lou. Her hands are lowered in the wide angle but raised in the closer view. (00:04:00)