Revealing mistake: In the shooting scene in the Mount Rushmore cafe, a boy in the background puts his fingers in his ears, because he knows the gun is about to be shot.
Visible crew/equipment: When the pilots are driving the plane and a spaceship pass them you can see a shadow of a microphone. (00:04:40)
Continuity mistake: Frederick picks up a bottle of champagne out of the ice bucket and shakes it a little. As he does this the foil and wire are still on top. He turns and aims the bottle at his wife, then tips it back and the foil has vanished as he now just removes the wire and cork.