During the introductory scene in David Green's room, when Van Kelt grabs McGivern from behind and administers a noogie, McGivern elbows Van Kelt and escapes his grasp, leaving his overcoat in Van Kelt's hands. He then darts out the room. In the next cut, a long shot from down the crowded hall, McGivern waves his overcoat over his head as he runs away from the room and Van Kelt chases him.
When the class asks David whether or not he trusts them to determine whether he or Dillon cheated, his sweater collar changes between shots.
At the homecoming football game, David first appears in a soiled jersey. When the game begins, it's clean.
After David's brawl in the alley, his face is bruised. This brown mark fades, then darkens between shots of him walking through the preparatory school for the first time.
When David picks up his father at work, he is driving the car. In close-ups, the car is moving when the camera is on David, but not when it's on his father.
Though it is a funny scene, there are only two ways that the boys could have been able to put Cleary's car in his room: Carrying it inside or disassembling the car outside and reassembling it in his room. Both are totally impossible, given the effort it would take for either method and the amount of time they had during Cleary's stroll.
David Green chooses to play football on Rosh Hashana, as opposed to observing the holiday. That evening, the scene cuts to David reciting "Avinu Malkenu" in the school chapel, when the principal walks in and reminds David that his holiday ended at sundown. In actuality, Rosh Hashana is a two-day holiday and would have extended through the following day. Assuming the football game was played on a Saturday, there is no possibility that this was the second day of Rosh Hashana, as Rosh Hashana never ends on a Saturday night (the two-day observance may only begin in the evening on a Sunday, Monday, Wednesday or Friday).
David is right-handed. From the receiver's point of view, the ball should rotate counter-clockwise when he throws it. At the end of the first game against Winchester, the last two shots of David's game-winning touchdown pass, from the receiver's point of view, the ball rotates clockwise.
When asked David Greene where he came from, he said, "Scranton, PA." Two-letter postal abbreviations for state names went into effect on July 1, 1963. This movie takes place in 1956, the state abbreviation would have been Penn.
When David picks his father up at the steel mill the car he is driving has a front license plate on it. Pennsylvania did not have front license plates in 1956 (as seen on a close up of the bus) or any time since.
When David Greene goes outside in the rain to wait for whoever made the sign with the swastika, the rain falls diagonally from two different directions.
When David is having dinner with the Dillons, Mr. Wheeler is introduced to Tom Keeting, who is wearing glasses. When he is about to sit, his glasses disappear. When he is sitting in chair, his glasses are on.
At the very beginning of the "Smokey Joe's Cafe" scene, actor Anthony Rapp is caught directly looking into the camera while playing drums on his pillow.
In the French class, two students do a "fist bump," which no one did back in the 1950's, after one of the students gets a high grade on an assignment or test. The fist bump came many years later.
When David picks up his father at the coal mine, and they're waiting for the train to go by, several cars are marked Penn Central and Conrail, which didn't exist in the late 1950s.
A 1965 Ford Mustang appears prominently In the Concours d'Elegance.
When David drives to the diner, the town has modern traffic lights.
In the dance in the school gym, one of the wall decorations is a Gone with the Wind (1939) poster designed for the 1960s re-release of that film.
Mr. Cleary uses the term "quantum leap" in its modern sense, meaning a large increase. Dictionaries put the first use of the term, as opposed to its original scientific sense, in the late 1950s or early 1960s. It's very unlikely that a French teacher in the 1950s would use it that way.