You could legally shoot an elk (or moose) in ID and/or MT with a .17 Hornet. I wouldn't try it, though I solidly believe 22 centerfires are fine for deer. Elk, I'd be on the fence till I saw it happen a few times.

I am pretty sure Scenar killed a big bull with a 220 Swift, but believe it was a target of opportunity for him while predator hunting, not elk hunting.