The tradition of leaving the bottom button of a coat undone started as a fashion choice. Coats are designed to flare away at the bottom, since the hips are (generally) broader than the waist, where a coat should be fastened. Modern suits are all cut with this in mind, and thus the bottom button should never be used, as it throws off the tailoring of the suit, turning a man’s silhouette into a cylinder.
Ideally, three-button suits aren’t supposed to have the top button fastened, either; leaving it undone balances the look and permits the lapel to roll closer to the waist, where it should be. In fact, the two-button suit takes care of this nicely by removing the offending button altogether. It is possible, if inadvisable, to button the top button without violating fashion rules.
There are a few exceptions. The bottom button on some double-breasted jacket is buttoned. There is also a type of custom jacket called a “paddock coat,” where the placement of the buttons is altered and both are buttoned. John F. Kennedy was known to wear such a jacket.
The custom of leaving the bottom button on a waistcoat (‘vest,” in American English) undone comes from the early 20th century. King Edward VII was too rotund to fasten his bottom button and the custom came from his imitators.