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: When Thomas goes by the level crossing, a close up of Thomas is shown but it's the same footage from season 1, only flipped. Because of this, Thomas' number is mirrored.