There are several chronological errors in the pictorial time-line shown during the musical title sequence. For example, the Moai of Easter Island are shown as "BC" and the Giza Pyramids follow the Chichen Itza 'El Castillo'.
In one episode, Sheldon states that pets are banned under the roommate agreement, but in another he adopts 25 cats. However, it's regularly shown that Sheldon uses the roommate agreement for his own advantage, often unaware or completely ignoring how it affects Leonard.
Whenever anyone is using the stairs in the apartment building, it is obvious that each floor is the same set, slightly redecorated. This can be most easily seen in the distinctive marks on the lamps that are next to each apartment doors (especially the one on the left.)