I can get the exact details, but I know that "Babes in Arms" opened in 1937, (which was still technically in the depression as America didn't fully get back to work until the war industries kicks-tarted the economy in 1942) and limped along for a few months until two other musicals closed, at which point IT WAS THE ONLY MUSICAL OPEN ON BROADWAY. Granted, there were a few plays, but only one musical. And it stayed that way for months. AND it turned Babes into a hit.
Also, during 1936, my beloved "On Your Toes" played for quite a while against only one musical, a revival of The Ziegfeld Follies.
One show.
The threads on BWW would be pretty thin if that happened here.