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. But it's regularly shown that Sheldon uses the roommate agreement to his own advantage, sometimes unaware, but more often completely ignoring, how it affects Leonard.
Howard states many times that he has a Masters degree in engineering from the Massachusetts Institute of Technology. However, it is also stated often that Howard never left his mother's home before he married. While MIT does have distance learning programs, Howard's degree program would have had required him to attend classes on campus, which would have required him to have moved out of his mother's house prior to the start of the series.
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.)