When the Russians are ready to leave they're shown in a group shot talking to Morticia and Gomez who is reading stock market tape coming out of the machine and the part of the tape that comes out of hangs straight down outside of the wire waste basket. After a couple of close up shots where Gomez is kissing Morticia's arm for speaking French they switch back to the group shot only this time the tape coming out of the machine is twisted and going into the basket.
When Thing pours the tea, part of a sleeve can be seen on his wrist.
When Mr. Harris drops the fish into the piranha tank, the prop fish can be seen in the bottom left corner of the tank after its "skeleton" has been thrown into the air by the vicious piranhas.
When Gomez blows up a 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).