None of the organizations that make up the Cat Fancy list a breed that is required to be both black and white (at least that I can find!). There are, however, breeds that fall on either end of the spectrum.
The photo above is of a Ragamuffin. The Ragamuffin cat breed began as a variant of the Ragdoll (although the CFA breed article indicates that the origins of the breed are unclear). Its full coat makes for a great looking black and white cat.
Some breeds, such as the British Shorthair, can be shown in what is called "smoke." A cat with the smoke pattern is a standard color, but instead of maintaining that color all the way along the hair shaft down to the roots, the undercoat is white (or silver).
When at rest, with the fur laid flat against the cat's body, the cat may appear to be a solid (self-colored) black cat. When the cat moves, or bends, however, the undercoat becomes visible, providing what's known as a shot-silk appearance. This gives more depth to the look of the cat's coat.
Famous Black And White Cats
There have been a few black and white cats that reached a certain level of fame.
Felix the Cat is one of the most recognizable characters in the world.
Felix's body is black, and I've seen him often referred to as a black cat, but his cheek and chin area are white.
Everyone knows Sylvester, the cartoon cat who chases Tweety bird around in the Looney Tunes cartoons.
As for real live black and white felines, White Heather was the black and white Persian who was Queen Victoria's favorite cat. When Queen Victoria (1819-1901) died, care of White Heather fell on her successor, King Edward VII (1841–1910).
Simon, pictured here, was a tuxedo cat who served aboard the HMS Amethyst and kept the rats under control, even after being wounded in battle.