My purple has faded really bad. It looks half-grey / half-light blue.
I'm weird and sentimental. I've worn it in a lot of comps etc.
It would also be nice to be able to dye it to brown when I get promoted.
Anybody?
It would be like painting your old beater instead of buying a new car. 
I did it with my karate belts when I was a kid (or my parents did, I should say) but never with BJJ. I suppose it's do-able.
I burned through my brown belt in likes year.
Just bought another one.....
We had a surplus of white belts back in the day. We dyed those. Never from one color to another. 
I washed and dried my Atama blue belt a few times after I first got it to shrink it a bit. It ended up being hard to tell if it was a really faded blue or really faded purple. I dyed it with some royal blue and it looks great now. Looks like the traditional light blue I wanted.
I rinsed the shit out of it multiple times because I was scared of ruining someone else's gi.
don't do it.
i dyed my beat up white belt blue... i visited a different school to train.
my belt looked like a REALLY faded purple... i had a room full of purples, half crazed blues, and a few browns eyeing me up...
i even had the brown ask me "in a round about nice way" who gave me my belt. i said caique, he's got this "the fuck?" look on his face because caique is legit... so he has this new look of "you fake ass motherfucker. you googled bjj and saw caique and made this shit up, i'm going to rape you on the mat even more now."
i figured out the weirdness... i explaiend i was a new blue and how i dyed it blue but it came out looking like a faded purple...
bros, it looked like i had spent 10 years at purple or some shit...
we all laughed about it later and i promptly spent the money to by a blue off the atama website.
i was literally promoted the week before. now imagine these blue belts w/ years in and the look of disgust on everyone's face.