The fur-bearing trout is often regarded as a (not-so) elaborate hoax along the lines of the jackalope. Unlike the doctored rabbit1 however, furry trout do exist - as a result of a fungal infection known as saprolegnia. Saprolegnia appears as a white, cotton-like growth which coats the skin and fins of fish. It generally only appears on areas that have been scratched, but occasionally will coat the entire body of the fish. Most healthy fish can survive a minor infection.
1. Egads! factgirl informs me that the jackalope is real, thanks to a nasty tumorous growth. What is the world coming to?