As was said early on in this thread, "it depends." What does it depend on?
1. The animal you're hunting. I like to hunt. To me, that means getting as close as is reasonably possible to the animal you are intending to shoot. If that means more busted stalks (and it does), that's part of the sport.
So, and this is just my perspective:
General plains game: I like to be under 100 yards, but I'm prepared to go to 200. I have made longer shots on plains game, but there's usually been a reason. I will admit that it wasn't always a good reason.
Dangerous game: I like to be under 50 yards, whether we're talking buffalo, lions or leopard. Elephant is even closer. The only one I shot was at 12 yards. The stalk was the hunt. The shot was just the end.
Specialty game: That depends on the animal and where you are. For bongo and forest sitatunga, I'd suggest anything over about 15 yards is too far. You aren't even likely to see them at that distance. Mountain nyala on the other hand can easily be over 300 yards, and I took a shot at 375 yards. That's the nature of shooting in the mountains.
2. Your skills and your equipment. Trying 200 yard shots for the first time on game is, in my view, unnecessary and irresponsible. If you want to take those shots, you should practice so that you can say up front that you are virtually certain to place a killing shot at the relevant distance. Equally, if you want to take those shots with a .416, you have the wrong equipment, so you better be extremely knowledgeable about ballistics. That's not what a .416 is for, but it is what a .300 Win Mag is for. so have the right equipment.
3. Whether you're hunting or target shooting. I think once you get beyond what most would consider a reasonable distance for the game you're after, then I'd suggest you're target shooting, not hunting. What's a reasonable distance for this distinction? We can argue about that. From my perspective, on general plains game, anything over about 250 yards is not really hunting, but for those who might have an issue with that, can we agree that wherever the line is, 400 yards and beyond exceeds it - even for those who have both the skills and the equipment?
And if you're target shooting rather than hunting, could I suggest that using live targets is unnecessary? It violates my personal hunting ethics, but I won't impose those on others. Having said that, I'd encourage those who engage in that sort of thing to consider whether Tannerite wouldn't provide a more satisfactory reaction.