I prefer to be slightly overgunned. As long as the recoil isn't troublesome, there's no real downside. Being slightly undergunned usually is no issue either, but when it is, it can be a real mess.
I don't have much use for the .223/22-250/220 Swift class for big game, but I do have a .228 Ackley which, with 90 grain bullets, has accounted for several mule deer bucks. I would be comfortable enough using it - or its neighbors like the Barnes .226 QT - on general big game.
On the whole, though, I think the .22 bore is best reserved for pests. Deer and pigs (and similarly sized animals) get a .25 at a minimum, and the faster the better. I don't really start to get comfortable until .27 territory, and don't start feeling silly until we get up to the .340 Weatherby or so. (I like the old .338 Winchester quite a bit. That level of recoil gets to me pretty quick, but as an all-around North American hunting cartridge, it's high on my list.)
Dangerous game? The .375 H&H is my minimum, and something starting with a "4" would make me a lot happier. And if my life really was in the balance, a .577 wouldn't be too big!