The broken gene's action is cumulative, meaning that two copies results in a greater impact on the rabbit's coat color than that of one copy of the broken gene and one of the non-broken gene. There are also a bunch of modifying genes that fine-tune just how the broken gene is expressed, which is how you can get a mostly-white rabbit that really has only one copy of the broken gene. It's a nuisance if you are trying for showable brokens, because (as others have said) the breed standards generally want more color in the coat.
"Charlies" got their name from the fact that the rabbits with two copies of the broken gene usually have very little in the way of a nose marking, if anything. Rather than a full butterfly, they usually have only a tiny mark near the nose, which reminded someone of the mustache of silent-film star Charlie Chaplin. The dark side of Charlies is that the broken gene also influences gut motility (the rate at which the digestive system processes food). Every rabbit with the broken gene has a digestive system that runs a bit slower than that of solid-colored rabbits. In a single-copy broken, the difference is slight, but the digestive systems of Charlies go a good bit slower. Any rabbit can get hit with GI stasis (basically, a digestive system that stops working entirely) but the tendency is much greater in Charlies. You can often spot the slow-running digestive action of a "true" Charlie by the irregular size and shape of the feces. Normal "bunny berries" are the same size, round, slightly flattened - rather like M & M's. A Charlie's poop is more like the peanut M & M's. With a rabbit like that, it is critical that they have plenty of fiber and water all the time, because they are more likely to get into digestive trouble than rabbits with normal digestive systems.