A fermion is a particle whose spin is an odd multiple of $\frac 1 2$, and a boson is a particle whose spin is an integer. From what I've seen, these appear to be the only two possibilities; not only have no particles been discovered with a spin of, say, $\frac 2 3$, it seems as though that's not even considered to be a possibility (in contrast, as I understand, there could be a spin $\frac 3 2$ particle, but it happens that none have ever been discovered).

Why is this the case? Or am I mistaken? Sorry if this question has been asked before, I couldn't find any like it.