Continuity mistake: In what is supposed to be a time-lapse scene, we see the Monsters University campus as nighttime ends and the sun rises. The shadows of the trees and buildings never change position as the sun gets higher in the sky.
Other mistake: Towards the beginning of the film, the teacher counts 19 students then says one is missing, implying there are 20 (including Mike) However, Mike ends up being paired with the teacher, despite there being an even number of students and therefore everyone should have a partner.
Deliberate mistake: In Monsters, Inc., when Mike and Sulley walked through Monstropolis to go to the company, Mike said to Sulley "You've been jealous of my good looks since the fourth grade". In this film, we're told Mike and Sulley met in college. [In the commentary, director Dan Scanlon explained that there was a lengthy scene that would have followed Mike and Sully throughout their years in elementary school, but it would have slowed the pace of the film and would have reduced the impact of their rivalry. Rather than spend a lot of time on a back story, Scanlon decided to have Mike and Sully meet in college, even though this was inaccurate.]