Factual error: Players didn't wear numbers on uniforms at that time.
Factual error: In Danny Kaye's song about "The Emperor's New Clothes" he persistently and repeatedly uses the words "the king is in the altogether." Hans Christian Andersen lived from 1805 to 1875, but the expression "in the altogether", meaning naked, was invented and popularised by George Du Maurier in his novel "Trilby" which was not published until 1894.
Revealing mistake: In the opening scene the Club's dancers are in full-swing, including the tall, lanky gentleman from Lautrec's famous Moulin Rouge poster, the one with the very angular, extended nose and chin. In several closeups it's very plain that the chin is a makeup appliance: the edge of the appliance isn't blended very well with the actor's real skin.