Morticia and the Psychiatrist - S1-E2
Continuity mistake: When Morticia is watching Gomez and the trains, at the second of the collision and explosion, it's a re-use of the shot when Mr. Hilliard is there, as in just a split second, you can see him, then there's Morticia again.
The Addams Family Goes to School - S1-E1
Continuity mistake: After Mr. Hilliard goes through gate and nears the house, it's really at the real house. A second later, he's approaching the now-familiar front porch on the studio set. This is easy to tell because the layout of the two porches are radically different.
The Addams Family Tree - S1-E5
Audio problem: With Morticia in his arms, he says "And I dropped you...only once." If you watch his profile, you can see that his face never moves. Hard to believe especially when you know he had an animated face.
Christmas with the Addams Family - S2-E15
Other mistake: As Morticia starts to play the flute, Gomez appears to have been filmed backwards, his hair and suit is reversed.
Christmas with the Addams Family - S2-E15
Revealing mistake: When Morticia is playing the flute, the strings that operate the basket lid and rope are visible.
The Addams Family Goes to School - S1-E1
Audio problem: When Wednesday comes into the house crying her mouth isn't moving.
The Addams Family Goes to School - S1-E1
Audio problem: After Gomez suggests the rack for a stretch, he rings for Lurch. You hear Lurch say "You rang?", but his lips don't move.
Cousin Itt Visits the Addams Family - S1-E20
Revealing mistake: In the wide shot of the family watching Cousin Itt perform magic tricks, you can see Felix Silla's legs as they probably didn't intend for the whole character to be in frame.
The Great Treasure Hunt - S2-E19
Revealing mistake: When Fester gives the anchor to Morticia, she drops it, and it bounces back up. When she lays it down, the wire used to achieve the gag is easily seen.
Visible crew/equipment: Whenever Thing comes out of his little places you can see a wire that is used to open up the little hiding place.
Continuity mistake: While playing chess, and when Pugsley calls him, Thing is a left hand. When he comes up in the garden area, he's a right hand.
Uncle Fester's Illness - S1-E18
Continuity mistake: When we see Wednesday playing a game with Thing she is missing two front teeth. Later when the doctor comes to check on Fester she is only missing one front tooth.
Continuity mistake: After the doorbell knob retracts into the frame, the door starts to open. After the angle change, the door starts to open again.
The Addams Family Goes to School - S1-E1
Continuity mistake: When Wednesday is first holding her doll, the buttons are facing forward. A second later, it's backwards.
The Addams Family Goes to School - S1-E1
Other mistake: When Thing is first seen, he's a right handed thing. When Wednesday gets the mail from him, Thing is a lefty.
The Addams Family Goes to School - S1-E1
Other mistake: As Gomez and Hilliard come to the top landing, the supposed access behind Gomez at the top of the stairs can be seen to be either a blank wall or a door that should be a hallway.
The Addams Family Goes to School - S1-E1
Visible crew/equipment: If you look at the shrubbery, you can see the person performing our first glimpse of Thing moving around.
Happy Birthday, Grandma Frump - S2-E22
Continuity mistake: As Morticia and Gomez play with the train, there is only one train on the set. When Gomez blows it up at the bridge, there is a second train in the midst of it. This is a reuse of stock footage.
Visible crew/equipment: As Thing withdraws into the tree, in the wide shot, you can see someone's elbow pop out from the tree.
Revealing mistake: As Fester looks at the mirror, it shatters. After he asks a question, it falls off the wall. After it does, you can see what appears to be someone's fingers pushing from the other side to knock the mirror down. That spot on the wall is obviously fabric covering a hole.
Answer: Basically from what I understand, they stuck the innards of a small flashlight into the base of the bulb and put a little switch on it. When he put it in his mouth and bit down, his teeth would hit the switch, which would turn on the flashlight and made the bulb light up.
TedStixon