Reference librarian Miss Costello (Joan Blondell) says, in response to a question about baseball star Ty Cobb, "He played only with the Detroit Tigers for 21 years." In fact, Cobb played major league baseball for 24 years - 22 years with the Tigers and two years with the Philadelphia Athletics.
Mike has just been made a vice president. When he closes the door to shut out the sound of the bongo drums, the bottom map behind him briefly changes to be of South America. For the rest of the scene the map is of Central America.
During the X-mas party scene in the reference room, Spencer Tracy and Kathryn Hepburn are sitting upstairs between the shelves of books. She is holding a paper cup of champagne in her right hand. Her right thumb is above the half way point on the cup, then suddenly, the same thumb is much closer to the bottom of the cup.
When Bunny Watson and Mr. Sumner are talking in the hallway shortly after they meet, she tells him that she doesn't smoke. But we see her starting to light a cigarette in her office, prompting Peg to comment that she only smokes when she is worried.
In the opening shot of the film, at Rockefeller Center, the shot begins at ground level and tilts up the building, but it was clearly shot from the top of the building down to ground level and then reversed because all the people on the ground are walking backwards.
Sumner explains to his tour that the has a handful of punch cards containing Shakespeare's play Hamlet. He is holding no more than 20 cards. Since the text of Hamlet contains over 160,000 characters (62,000 if compressed) it would be physically impossible to store that much data on so few punch cards.