Factual error: The baby's pacifier that was used was not made at that time. It looked more like a NUK than the old rounded rubber tip.
Factual error: Mid movie, they show a Timex wall clock. But Timex did not make synchronized wall clocks. Simplex did.
Factual error: During the car chase, a rear window brake light is visible in John's car as he looks back at his pursuers. These devices weren't invented until the 1980s.
Factual error: At the Nobel Prize Ceremony, Nash is giving a speech. During a wide-shot right before he leaves the stage, you can see a few girls wearing white military style caps (to the right of the screen). Hats are in fact worn by Swedish students at the Nobel Prize Ceremony, however the style is completely different. The caps should have a much smaller, soft crown and not the wide crown of a military hat.
Factual error: John Nash enters Princeton as a graduate student (Ph.D.) candidate in 1947. By that time the Graduate College had already been built (1913) and is about three miles from the main campus. At that time, all graduate students had to live at the Graduate College building, and not where Nash is shown living with his imaginary roommate at the center of campus.