Factual error: The main apartment in the show is located in Pasadena, Calif. When looking out the window in the background, mountains are shown in the distance and far away. In reality, Pasadena lies right underneath the long mountain range so the mountains would be up close, not far away.
Deliberate mistake: Very often, during Skype or other video chat conversations, the people on the computer screen look left or right when talking to someone who is not directly in front of the screen. Although it certainly helps the viewers understand who's talking to whom (which is why I assume it's a 'deliberate mistake'), it isn't realistic, since it implies that the people on the screen are actually constantly looking away from their computer instead of at their screen or webcam.
Revealing mistake: If you look at the marble patterns on the walls, they are all exactly the same from floor to floor. Also the bricks oddities on the stairway match from floor to floor. They change lights, elevator tape etc. but it's the same floor used over and over as different floors.terry s
Continuity mistake: In the beginning of the series Sheldon knocks on Penny's door only once. Later in the series Sheldon explains he has always knocked 3 times because he walked in on his father with another woman.