Continuity mistake: In the first minute of the film before the title appears, we see the Medici Brothers Circus train leave Sarasota, Florida and head north into Georgia, touring the southern United States before arriving in Joplin, Missouri (where the first half of the film takes place). Upon arrival, the train's engine itself comes to a full stop directly in front of the Joplin depot. This shot immediately cuts to a frontal shot of the train engine coming to a full stop in open countryside with no sign of the depot or civilization anywhere, except for a fleeting glimpse of a rooftop about a hundred yards in the background.
Other mistake: The titular baby elephant gets his name from the sign on his cot malfunctioning - the "D" in "Our Dear Jumbo" flips over onto the second line, replacing the "J." The problem is that as shot, there's no way for that to happen - either the letter's glued on and should have fallen off entirely, or it's screwed on and became loose, rotating around the bottom screw - in that case the D would be backwards.
Factual error: The silhouette elephant weather-vane is incorrectly made and would never point into the wind because the elephants are too far forward.