Revealing mistake: Pay close attention to some of the scenes on the airplane during the flight that the Walshes are taking to Hawaii. Several times you will see a large, winding (and ascending) staircase in the background of the plane. If you also look close, you will see an 'exit' sign' slightly behind and to the left of the staircase. While some luxury airbuses do have staircases, given the fact you can see an exit sign it is easy to tell these scenes were shot in some kind of setting other than an airplane.
Revealing mistake: At the end of the episode in the Peach Pit, Nat and the gang are all singing Christmas songs while Nat is playing some of the songs on a keyboard. It is very easy to tell by looking at his hands, Nat is simply miming on the keyboard.