In the passage-of-time montage, a newspaper, supposedly from January 2, 1946, a headline reads "Buckeyes Rout USC in Annual Rose Bowl Game". Actually, Alabama beat USC 34-14 in the game that year. However, the Ohio State Buckeyes did beat USC in the 1955 Rose Bowl, 20-7.
During the "March of Times" type sequence, it has Dwight D. Eisenhower being inaugurated in 1952. Inaugurals are in January the year following election.
Early in their reunion, where the guys are sitting together and awkwardly fumbling for something to talk about, one asks the others who they think will win the World Series. The 1955 Series actually ended on 4 October, a week before the events in this film. (The Brooklyn Dodgers beat the New York Yankees four games to three.) It should have been obvious to the filmmakers that the World Series of that era would have ended well before 11 October, no matter who was playing or how many games it went.
During the scene in the taxi just after Ted Riley (Gene Kelly) mentions Kid Mariachi to Jackie Leighton (Cyd Charisse), the street scene in the rear window of the taxi goes blank for about six frames.
After Ted gets his "Dear John" letter, he and his buddies leave Tim's bar (where they were the only people at the bar) and go to at least 7 other bars where they are, against all odds, the only people at each one. Then following a dance routine, they end up back at Tim's where, again, they are the only customers in the establishment. However, 10 years later, Tim's is standing room only at noon.
In the 1945-1955 montage, the shot of the 1953 New Years Eve crowds at Times Square shows Sundown (1941) at the Criterion Theatre, so it's obviously New Years Eve 1941-1942 footage.
In the opening montage of the three soldiers returning home in 1945 after the war, there is a shot of an ocean liner in New York harbor. The ship is the Italian liner Andrea Doria, which did not make its maiden voyage until 1953 (and which sank after a collision in 1956). Also, if the Doria is meant to be the ship the men are returning home on, it is facing the wrong way, headed out of the harbor instead of into it.
When Ted runs out of Tim's Bar after reading the "Dear John" letter from his girlfriend, at least one of the cars on the street (a taxi) is an early 1950s model, although the scene is set in 1945. The taxis most prominently featured are 1946 DeSoto models, a popular post-war style, but which were not yet in use on the streets as early as October 1945.
At the reunion at Tim's bar, after the boys all drink a shot, Angie says, "If Connie could only see me now, drinking in the middle of the afternoon." The meeting was set for noon, so at most it's 5-10 minutes past 12; hardly the "middle of the afternoon".
When Jackie comes to Stillman's gym to see Ted, Butch describes her as "about five foot five". Cyd Charisse is 5' 7½" (plus her slight heels); while Phil Arnold (who plays Butch) is only 5' 1", being in the fight game, he should have been able to come up with a closer estimate on Jackie's height.