Among other things, the movie celebrates the 1933 repeal of Prohibition in the USA by making every character (especially Powell’s character, Nick Charles) drink copious amounts of alcohol at any opportunity. One of the drinks of choice is the Martini cocktail. According to Wikipedia:
The martini is a cocktail made with gin and vermouth and garnished with an olive. Over the years, the martini has become one of the most well-known mixed alcoholic beverages. H. L. Mencken once called the martini “the only American invention as perfect as the sonnet”, and E. B. White called it “the elixir of quietude”.
In the movie of The Thin Man, Nick Charles indicates the most important thing in making a Martini:
“Always have rhythm in your shaking. Now a Manhattan you shake to fox-trot time, a Bronx to two-step time, a dry martini you always shake to waltz time.”
When Nick’s wife Nora (played by Myrna Loy) meets Nick a few minutes later, she asks:
Nora: Say, how many drinks have you had?
Nick: This will make six martinis.
Nora (to waiter): All right. Will you bring me five more martinis, Leo? And line them right up here.
Which leads inexorably to today’s clerihew.