When Astaire, Wynn, and Skelton are talking about the play never being produced, there are several pictures of M.G.M. stars on the wall. First, from left to right, is Judy Garland, then Spencer Tracy, then Katharine Hepburn, and another I could not identify. The time period in the movie was at that point during the mid 1930s, but those photos were all from the late 1940s.
When Harry hurts his hand playing baseball with the Washington Senators, his glove is knocked off of his left hand and he holds the index finger on that hand in obvious pain. Immediately afterward, when he is having difficulty playing the piano because of his injury, he has a bandage on his right index finger.