Peggy announces she's leaving on Friday, September 13th because that's the day Monty promised they could be married. However, in 1944, September 13th was a Wednesday. They may have gotten mixed up with Monty's birthday being on October 13th, which was actually a Friday in 1944.
At one point, the desk calendar shows a full moon on August 22, 1944. Full moon was on August 4th that year.
When a phone rings on Brewster's desk, he picks up the wrong phone. His friend answers the ringing phone and passes it to Brewster and then places Brewster's hand set on the ringing phone's cradle, which would have ended the phone call. Brewster finishes his call and puts the handset on the other cradle. Then that phone rings and he has another conversation which is also impossible.