Let's assume that all these particles are identical. Now let's see what happens when we interchange two of them. I'll let you calculate it. If this number equals one has bosons, if it's one has fermions, but otherwise one has anyons . One can have fun playing with this idea.
Many people have.
Wilczek is a big proponent of an anyonic theory of high-temperature superconductivity, although recent experiments, demonstrating an apparent absense of spontaneous parity violation that one would expect in this theory, seem to rule it out. Something that anyone who has followed me so far can have fun doing, is to see what sort of particle a bound state of anyons acts like. The high-handed manner in which I've thrown out the symmetric group and started working with braid group statistics should disturb you, but again I can cite fancy mathematical physics papers which should allay your fears. Nonetheless it's all part of one story.