I’ve heard the expression, “My dance card is full,” and knew from context that it meant someone was busy. I never really thought about the etymology of the phrase though. But today I was listening to a Librivox recording of Richard Marsh’s 1897 horror novel The Beetle and I recognized the source of the expression.
In olden times, at formal parties, people would schedule dances with each other. A person would have a card or booklet listing all the tunes that were to be played, in order, with a space to write in the names of dance partners. The Wikipedia entry has images of some cards.

Programme du Bal, from http://www.ephemera-society.org.uk/items/2009/aug09.html
This fascinates me because it is so alien to my experience, yet the expression is something I have known for a long time.