After over 25 years hunting Wyoming as a nonresident our family has taken 2 sheep, a moose, about a dozen mule deer, a couple elk and a couple dozen buck antelope. If my memory serves me correctly, we've had one tag go unfilled, which would have easily been filled with one more day of hunting. I can say I've never had a bad experience in that state. The wardens and biologists have consistently been some of the best and most helpful people I have worked with.
We apply in 8 western states each year, and there are very, very few hidden secrets on limited entry hunts. There are too many applicants, magazines and info services covering these hunts for there to be many "undiscovered" hunts. Easy draw odds almost always equate with a difficult hunting situation....either physically difficult, limited public land, or low hunter success.