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.
Deliberate mistake: In the PBS airings in the U.S. of "Tuneful Toots" and "Duncan and the Old Mine", Rusty is referred to as a female rather than a male.