Whether you think of it as a traditional New England "boiled dinner" or an annual St. Patrick's Day treat, the combination of simmered corned beef and cabbage -- usually with potatoes and carrots as an accompaniment -- is a natural pairing. The rich and briny flavor of the meat permeates the vegetables, while the vegetables mellow the flavor of the corned beef. However, if you have picky eaters or just want to keep the broth clear for soup, you can separate the meat from the vegetables.
Corned beef is made from the belly meat of a steer, a thick slab of leather-tough meat with long strands of muscle fibers that give it a pronounced grain. For centuries cooks have known that salting this piece of meat helped tenderize it, as well as preserving it, and produced a piece of beef that rivaled hams and bacon for its rich flavor. Depending how heavily the beef was salted it might need to be soaked or par-cooked in several changes of water. Once it's finally cooked for the table, the corned beef is just salty enough to produce a well-seasoned and richly flavored broth.
Centuries of frugal cooks have considered that broth to be a culinary asset, and have taken advantage of its flavors by cooking the rest of the meal in the same pot with the corned beef. Traditionally, this consisted of inexpensive long-storage vegetables, including potatoes, carrots, rutabagas and, above all, cabbage. The cabbage provided both a healthy quantity of fiber and some vitamin C, both valued by the end of a long, cold winter. The combination of corned beef and vegetables came to be known in New England as "boiled dinner," and it's still a much-loved tradition in New England. The leftover broth, well-fortified by the vegetables, made an excellent base for the next day's soup.
If you'd rather cook your corned beef and vegetables separately, there's no reason not to. Often it's easier or more practical to cook each vegetable separately to its best degree of doneness, rather than cooking them all together in the same pot. It's also handy if you don't want your beef or its broth to taste of cabbage, or if you have picky eaters who object to one or another part of the meal. If you'd like your potatoes or cabbage to have the traditional flavor, you can always add part of the beef broth to the respective pots while they're cooking.
Part of the appeal of a traditional boiled dinner is the leftovers, and cooking the vegetables separately doesn't need to change that. For example, you can still dice up the leftover beef and vegetables to make hash. Add a half cup of reserved cooking liquid to the skillet and let it simmer, lending its flavors to the potatoes and vegetables. Then add oil or butter, and let the mixture brown as you usually would. The reserved beef broth, free of the potatoes' cloudy starch, makes a fine soup with navy beans or split peas and the diced leftover vegetables.