Rich's answer stands: what do you mean by "Big Game"? In Africa and parts of Asia, North American deer and elk would barely class as "medium game." There are a lot of hunters here who hunt BIG GAME. (Big browns, buffalo of all kinds, African lion, elephant, hippo, Alaskan moose, etc)
Jack O’Connor "proved" that you can effectively use the .270 for just about anything. That doesn't mean that it's the RIGHT choice for Polar Bear, but he managed to do it. With the heavier bullets, I'd feel fine with anything in the Americas with it, with the possible exception of big browns. (But with the right shot, maybe even then, though it's not optimal.)