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.
Uncle Fester's Toupee - S1-E31
Visible crew/equipment: As Gomez goes to ring for Lurch, the shadow of the boom mike is on the wall behind him.
Uncle Fester's Toupee - S1-E31
Other mistake: Bad editing shows up a couple times in this episode. Once when Max goes to ring the doorbell. As he reaches for it, it moves out and in on its own and he moves with it. As he does, its easy to see the film is run back and forth. The second time is when he reacts to the news of Kitty. He flinches, and his toupee slips over his face, then the film is run backwards so it goes back.
Crisis in the Addams Family - S1-E24
Visible crew/equipment: In the play room, as Fester goes to leave, the shadow of the boom mike is seen on the wall behind him.
The Addams Family Meets the Undercover Man - S1-E16
Continuity mistake: After Conkey says "Yes, it is", in the long shot, he lowers his arm. As he goes to answer Gomez, his arm is suddenly up.
The Addams Family Meets the Undercover Man - S1-E16
Audio problem: Ted Cassidy (Lurch) was an accomplished organist. However, when he "played" the Harpsichord, he was only miming the actions. When Lurch is playing while Gomez and Morticia are fencing the music doesn't match the movements of his hands.
Wednesday Leaves Home - S1-E10
Other mistake: As Fester comes into the room where Gomez and Morticia are, the piece of the cactus leaf that Fester is to eat is differently colored than the rest of the leaf.
The New Neighbors Meet the Addams Family - S1-E9
Revealing mistake: As the Petersons look at the square of carpeting coming up, the wire pulling the door up is visible.
Other mistake: As Fester goes to shoot Gomez, his aim strays upward to about a 45 to 50° angle. Not enough to shoot the support chain on the light that is directly above him, and hits him on the head.
Continuity mistake: As Fester gets a pistol from the drawer, there is an assortment of items in the drawer, and he leaves it open. A moment later, he goes to exchange guns, the drawer is closed and suddenly bare, except for the other pistol.
Continuity mistake: As Fester talks with Gomez, he has a fork in hand that has a small chunk of meat on it. In one quick take, there is an entire meatball on it, then a moment later, it's a small bite again.
Morticia Joins the Ladies League - S1-E6
Continuity mistake: As Pugsley comes out of the closet counting his money, he's dropping coins from his right hand to his left. A second later, it's reversed, and his facial expression changes in the same moment.
Continuity mistake: After talking to Cousin It, Morticia is reading a book titled "The Law of Bigamy." As the camera does a close up on the book, the label on the spine says "Homer's Iliad".
Revealing mistake: In the wide shots of the car, the camera side of the sound stage is reflected in the car's finish.
Visible crew/equipment: As Mr. Hinson and his aide get out of the car, the shadow of the boom mike is visible on the house.
Morticia, the Sculptress - S2-E9
Continuity mistake: As Morticia tries to sculpt Lurch, the boulder she is working on gets a sudden vertical crack in it from the wide shot to the close up.
Halloween - Addams Style - S2-E7
Continuity mistake: As Gomez and Fester bob for apples, the apples in Gomez's bucket alternate between swirling around and being still between close up and distance shots.
The Addams Family Meet the VIPs - S1-E11
Revealing mistake: As Morticia and Gomez throw the knives at the target, and they stack tip to hilt, we can see the wire they slide on.
The Addams Family Meet the VIPs - S1-E11
Revealing mistake: After Mr. Harris leaves the play room, Fester goes to lay down on the bed of nails. As he does, the squared outline of the board under his robe he lays on is visible.
Morticia and the Psychiatrist - S1-E2
Visible crew/equipment: When Uncle Fester climbs down the tree (after observing Pugsley interact with the shrink, a shadow of the moving dolly mic is visible as it moves backward. (00:22:35)
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