Factual error: When Midge is moving to her parents' apartment there is a stack of books on the table while her son is watching TV. On top is a Nancy Drew book (the back of the book which is yellow). However publisher didn't introduce the editions with the yellow back covers until the 1960s.
Factual error: Abe hilariously exercises in a ridiculous suit while listening to Robert Preston's "Chicken Fat" recording. But Preston (who starred in The Music Man) didn't release this until 1962, at the behest of President Kennedy, who was big on physical fitness.
Continuity mistake: When Midge descends the hotel's grand staircase, the piano player and guests are below. Once down the stairs, Midge finds Abe and Rose sitting in chairs with their luggage, just to the right of the piano, in front of a large pillar, with a round sofa behind. However, when Midge was walking down the stairs, the lobby looks different. The piano is in the same position relative to the stairs, but her parents and the chairs are missing and the round sofa is a different style and the shape of the carpeting in front of the pillar is also different.
Deliberate mistake: In a flashback to the Maisels' wedding reception (about 1953) Midge surprises Joel by having hired professional dancers, who are shown performing the well-known "Bottle Dance" (the part where they slide on their knees). This dance routine, which is often performed at Jewish weddings and other social events, was actually choreographed by Jerome Robbins for the musical, "Fiddler on the Roof," which debuted on Broadway in 1964. The dance did not exist at the time of Joel and Midge's wedding, but, as an inside joke, while Midge watches the dancers, she says, "That's really good. They should do something with that."