I ran across the old chestnut about various belt colors from white to black resulting from a white belt that gradually gets dirtier and dirtier over time on a discussion board recently. There are a couple of problems with this story, the biggest one being that it has no basis in fact.
Let's start with the fact that the traditional, white dogi tied with a belt isn't really traditional clothing for budo training in Japan. Traditionally hakama were worn for budo practice. There is no belt visible on the outside of hakama. The hakama is tied over the top of the obi. The modern training dogi was invented by Jigoro Kano, the founder of Judo, over a period of years at the end of the 19th century. It's based on classical Japanese underwear! In other words, it's not really traditional.
Kano Sensei also invented the modern rank system using "dan" ranks instead of the tradition teaching licenses that are issued by schools that predate the creation of Kodokan Judo. As Kodokan Judo grew into a nationally practiced martial art, he wanted a visible means of telling the difference between beginners and students who had the basics, and so they used white belts for beginners and black belts for experienced students. Originally there were also only 3 dan ranks, not the 10 that are now used.
As for the belt getting dirty over time, that ignores one great feature of Japanese culture. The Japanese are fanatically clean as whole, and the idea of letting a piece of your training uniform get so dirty and nasty that it turned black is ridiculous. No teacher would have let a student train with a belt like that, and no other student would have wanted to train with someone wearing anything close to that dirty. Students take pride in their uniforms and are expected to make sure they are clean for every practice. That includes the belt.
So, given the above, can we please put that dirty, smelly, obnoxious old story about dirty belts to rest? Please.