The Cuban sandwich was originated by factory workers who weren't allowed much of a lunch break, if any. The workers could eat their big, filling lunch with one hand while continuing to roll cigars or whatever with their other hand.
Cuban sandwiches (often referred to as just "Cubans") are taken very seriously in both Miami and Tampa, with the alternative newspapers doing annual taste tests of the best and worst Cubans.
In many restaurants and delis where Cubans are available, there are only two options when you order: with or without lettuce and tomato, and pressed or unpressed. Any other special request, such as "hold the pickles," will at best be met with grudging acceptance and at worst with outright hostility.
However, the most essential part of the Cuban is the bread. The same ingredients on a different kind of bread result in something other than a Cuban; most dining establishments then refer to it as a media noche (midnight sandwich), most commonly served on a sweet roll. Media noches are actually more readily available throughout the United States than Cuban sandwiches, since fresh Cuban bread is only available in cities with a large enough Cuban community to support a Cuban bakery.