Indiana Jones: So what are you, some kind of... triple agent?
Mac: Nah. I just lied about being a double.
The young Henry nicknamed himself "Mutt, which is also a term for a dog. Indy named himself after a dog he had as a child that was called Indiana (as noted in the third movie). This is the first hint about their relationship. See more...
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Indiana Jones and the Kingdom of the Crystal Skull (2008) - 137 mistakes
Other: When Indy escapes on the rocket powered cart, it is slowed down by a pool of water on the tracks. The only problem is that when this happens in real life, it's almost the same as slamming into a wall, and therefore Indy should have been flying several feet away from the rocket powered cart. You can actually see the chair that the test persons before him used, and it has double seat belts on it.
Continuity: Right at the start when the Russians pull Mac out of the car, they throw Indy's hat on the ground and there is a close-up shot of it where you can see Mac's feet right next to it. It then cuts to a relatively wide bird's-eye shot over the car as it is surrounded by Russian soldiers and there is nobody near the hat - Mac isn't even in the shot.
Factual error: In the warehouse when Indy and the Soviets open the crate with the alien's body, all the overhead lights are pulled towards it. If the magnetic field was that strong, the lights also would have been pulled towards the box before it was opened. (Put a toy car on top of a table and pull it with a magnet from under the table to see the field in action).
Continuity: When Indy is held captive in the Soviet's tent, Mac enters and they have a conversation. Indy then turns his head to his left, looking straight at Mac. Just before Indy makes his threat about what he'll do to Mac when he gets loose, there is an angle change, and Indy is suddenly staring straight ahead, with no time to turn his head in between shots. He then turns his head back to look Mac in the face again.
Continuity: Throughout the whole movie, Irina's hair seems to be kept absolutely pristine, even when she is flung from cars. This is very noticible at the car chase in the military base when she is thrown from the car by Indy. One minute, a few strands of hair are slightly messed, then they are fixed again. It is likely, however, given the many other nods to films of that period, that the "concrete hair" effect is a deliberate decision.