When Fester blows open the painting to access the safe, the safe's door is already hanging off its hinges before the painting even detaches from the wall. Also, the smoke from the explosion clears much too quickly.
When Thing feeds Cleopatra, he is a left hand, instead of his usual right. He is also a left when he hands Gomez a yacht circular, when he jumps in on the bidding for the stocks, and when he makes instant wine.
When Gomez blows up his last toy train, Mr. Hilliard can be seen for just a second standing next to Gomez, in stock footage from The Addams Family Goes to School (1964).