Carbon arc lighting was used in movie theater projectors, and had to be changed every hour or so. Therefore the projectionist in the movie would not have been able to keep the screen lit for longer than an hour because the carbon rods would burn out. In "reality" the movie would have ended before the missing character returned.
Movies are made in reels of approximately ten minutes each and supplied to theatres in spools (two reels joined together). So after a maximum of twenty minutes the projector would run out of film and show a white light.
Although the film's fictional namesake has scenes filmed in the Copacabana club in New York, the club was not open until November 1940. This is not consistent with the actual film, which is set during the Great Depression.
When Cecilia is playing the ukulele in the music store she is strumming along to the song and the song stops. She continues to strum along after the song is over, but there is no sound. The ukulele playing was obviously dubbed in.
As Cecilia and Gil play and sing in the music store, the camera casts a shadow in the lower left that disappears as it pulls back to a wide-angle shot.