If you're asking about why there's an upper limit to the spin, this Quora thread had some good info (Simmons-Duffin is a top notch expert on field theory): https://www.quora.com/Why-are-there-no-spin-3-2-or-higher-fu... Short answer is that for too high a spin the theories become non-renormalizable and break down [i.e. require additional physics to enter] going to ever shorter length scales.

If you're asking why it goes in steps of 1/2 it's a group representation theory thing (guessing you probably already know about that).