Continuity mistake: When John Nash enters the cafe at the end and the people start giving him the pens, the first two go down pretty much perpendicular to him but later it shows them all turned 90 degrees.
Other mistake: When Nash is put into the car and is being taken away to the hospital, there is a line of people watching him drive away. Two of the spectators are the imaginary roommate and the little girl. They are standing in front of rest of the spectators. As the car is driving away you see a female student look around the imaginary roommate to see the moving car. If the roommate can only be seen by Nash why did she need to look around the roommate to watch the car drive away?
Factual error: In a scene that takes place in 1956 or 1957, Alicia Nash places an orange Tupperware container in the refrigerator. Although Tupperware first became very popular in the mid-50s, the particular model of Tupperware used in the film was not introduced until the late 60s or early 70s.
Other mistake: In the end, when Nash gets standing ovations from the Nobel-prize audience, you can clearly see that the name Nobel is spelled wrong on the platform which he stands behind.
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: When John Nash returns from the Pentagon to his place of employment, he walks by an American flag. The flag has the staggered row arrangement of the current 50 stars, not the 48 star flag you should see in the 1950s.
Continuity mistake: When John's wife is in the bathroom she drinks all of the water in a cup, but when she throws it the cup is full again.
Factual error: When Alicia is wheeling her baby in front of Nassau Hall on the Princeton Campus and talking to Nash's former colleague, the building's handicapped access ramp is clearly visible in the background - not in those days...
Continuity mistake: In one of the first scenes where Nash is being introduced to everyone around the punchbowl, the trees around them have CGI leaves added on since it was shot in January and had to look like early fall. In shots that show the trees' shadows on building walls, all we see is branches. Shouldn't there be leaves?
Continuity mistake: Near the end of the movie, when Prof. Nash goes back to Princeton, he is wearing a checkered brown and green shirt. The next day, he is wearing a light blue shirt, but when the scene changes, he is wearing the brown with green shirt again.
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: Mid movie, they show a Timex wall clock. But Timex did not make synchronized wall clocks. Simplex did.
Continuity mistake: When the student approaches Prof. Nash in the Princeton library, he tells Prof. Nash he has been working on a mathematical theory. When the student hands Prof. Nash what appears to be a binder he does so by holding the binder with his left hand, palm down and fingers on top of the binder. In the immediately subsequent shot of both Prof. Nash and the student, Prof. Nash takes the binder from the student who is now holding the binder palm up, fingers under the binder.
Revealing mistake: The scene in the sanitarium where Nash is given the insulin shot (usually done with a dummy retracting needle) you can see the tip of the needle slide along the skin as it is supposedly withdrawn.
Deliberate mistake: Winners of the Nobel prize don't give an acceptance speech - they usually give a lecture before the award is given, and then at the most say "thank you," or in Nash's case "money would be nicer".
Factual error: When Nash and his fellow student are playing Go in the Princeton University commons, Cannon Green, you can see two tiger sculptures beyond them. These sculptures were not added to the campus until 1969 when Princeton became a co-educational (previously, it was all men) institution.
Continuity mistake: When Alicia puts some pills on the table/desk for John Nash, the pills move/disappear/reappear, depending on which shot is used.
Other mistake: When Nash receives his Nobel price, you can see the words "Stockholm, Sweden" at the bottom of the screen but in the French version the words are "Stockholm, Suisse" that means "Stockholm, Switzerland," as "Suisse" is the French word for Switzerland.
Continuity mistake: The second time Nash sees his old roommate's niece he drops his briefcase to hug her. A couple of shots later as he is walking away, his briefcase is back in his hands, without him having picked it up. Later in the film we see him from another man's point of view, and he is kneeling down, stroking the face of a girl who isn't there. He moves like he's really interacting with the imaginary people, so it can't be that he never really dropped his case.