Continuity mistake: In the scene where Bob gives Elvin the apple cider and another jar, when you first see the two jars, the jar in his left hand has a few inches of foam on the top. Bob sets the jars on the table, and the foam is gone. When the shot returns to the table the foam is back again.
Deliberate mistake: In the very first episode, the kids are fighting and arguing all throughout the house. Claire then asks Cliff, "Why did we have 4 kids?", referring to Denise, Theo, Rudy, and Vanessa. In all of the following episodes, they actually have 5 kids. I guess they decided to add Sandra (the oldest Huxtable) later down the road.
Other mistake: The outside shots show that the house is in the middle of a row of houses, connected to other houses on both sides. But in the living room, behind the stairs, is a window looking outside. There couldn't possibly be a window there as that is where the house shares a wall with the house next door.
Plot hole: When Theo is complaining that his father is inviting too many people to his graduation, he says "I saw him in the backyard 5 minutes ago hosing down the beach chairs", but this couldn't have been possible because in fact, Cliff was upstairs right before that scene, and then went straight to the living room, so there's no possible way that Cliff could have been in the yard at the time Theo had said he was.
Continuity mistake: Towards the end of the episode, Cliff Huxtable is sitting in the living room with his father Russell, Sgt. Stokes, and Homer. He is telling them of an incident that happened while he was in Bermuda. He is using dominoes to explain his story. At first there are about six or seven dominoes standing up on their sides. The camera angle switches, and all the dominoes are flat on the table. It goes on for a minute or so with the dominoes lying flat on the table. Then when Russell, Sgt. Stokes, and Homer get up to go for a walk, the dominoes are standing up again.