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.mirtom