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: Several shots have been taken from Series 1, including the group of guards firing in the hangar bay, and various shots of the shuttles during the chase.
Plane Death - S1-E14
Deliberate mistake: When the bad guy goes to smash a table onto Michael Landon notice the plate of glass in front of him, visible. The glass is there so Landon doesn't get hurt and to make the table break into pieces in front of him.
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.