I'm comfortable shooting at pretty much any distance that I've already verified on a range. Ballistic math isn't so difficult. Earlier this year I killed a klippie at 230 yards with a solid from a suppressed .375 H&H and that's a pretty small target. With smaller calibers (.264-.308) I've killed multiple animals beyond 500 yards and have recovered every one of them within a few feet of where they were shot. This has WAY more to do with experience, the wind, rest, technique, verified DOPE data, and the scope than it does with the caliber.
If you've shot uphill, on a windy, rainy day at 600 yards from an improvised rest you know what you and your rifle can do in those conditions. If you haven't, you don't. If you know from experience that you can reliably put every shot on target at a certain range......... well, then you know.
With a .375 H&H I'd be comfortable out to 400 yards with the right rest, scope, etc. Maybe out to 600 or 700 if it was to anchor a wounded animal. It's just as easy to dial 30 clicks on the scope as it is 15. It takes meticulous setup and trigger technique. A .416 Rigby isn't so forgiving. It's harder to shoot well from a solid rest and very few of them are fitted with scopes suitable for long range shooting. Some of them have enough inherent accuracy to shoot sub MOA, but probably most do not. People don't really tune them for supreme accuracy at distance because that's not their intended purpose. A .416 Rigby carries enough energy, but not really enough velocity for reliable bullet expansion (if that's important) beyond 200-250 yards. The .375 H&H isn't much better.
I have no experience with the 9.3x62.