I can definitely attest as to how important it is to leave grass at the 4" mark. As most people know we have gone through an extreme drought down here in the south. Those that pulled their cattle early and left there pastures fallow have recovered nicely. Those that opened their gates for free roam grazing still have no grass as it was killed off by grazing to the ground.
Here is a link to a very good article.
http://learningstore.uwex.edu/assets/pdfs/A3529.pdf
Jim; I will agree that the 4" mark, in general, is good, but there are some stubble grasses that benefit from grazing down to only 6 or 8".
There has been remarks made about grazing pasture during muddy conditions. Actually there are some grasses and pastures that may benefit from this. I have grazed Tifton 85 until the paddocks looked like mud pits. They recovered better than they were before. Organic matter was pushed into the soil via hoof action, making for better tilth, and rooting nodes were also forced into the soil making for a thicker stand. With other grasses and different dirt, one may only ruin the pasture.