When the Ballplayer is at the bar next to a wall calendar, in the medium shots the calendar page is for June 1954, but in a close-up the page says March 1954. The mistake would seem to be the close-up because there is a night-time baseball game on the bar's television and the baseball season didn't begin until April.
The movie takes place in 1954. In one scene, a man is in a bar watching the World Series. The TV announcer for the game starts talking about "Campanella, Hodges, and Berra" being involved in a play. In 1954 those 3 players played for Brooklyn Dodgers and NY Yankees, however the 1954 World Series was between the Cleveland Indians and NY Giants, therefore none of those 3 even played.