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: During the scene where the twin sisters were singing their song, Theo was asked by Cliff to go to Peter's house and ask his mother if he could stay. Theo left immediately after this. But if you look closely while the twins are singing, you'll see Theo's arm in the scene. When the twins finish performing, they sit down. After this, you see Theo coming through the door after having just gone to Peter's house.
Continuity mistake: The amount of water in Clair's cup changes a couple of times, most noticeably after Cliff tells her what she is eating, she takes a drink and the cup is about 1/3 full. She gets up to go find Theo, and now there is very little water in the glass.