I have seen many animals taken with "too much gun."
Most of the time the meat damage is no worse or even less than with normal "deer" cartridges ie. .270, 30-06 etc.
The bullets in the bigger stuff are made to be used on bigger game, they are tougher, they don't send pieces flying or blow a shoulder off the animal.
My dad has killed a couple pronghorn antelope with a 375 H&H with 270 grain bullets leaving the muzzle at about 2600 fps, you could almost eat to the bullet hole, and they were just as dead as the ones my brothers and I shot with our 270's and "06"s.
If you have a 338 you are covered very well for all north american game, don"t worry about being over gunned.
The biggest drawback is a lot of people don't shoot the bigger stuff as well, if you can handle the recoil and shoot it well you have nothing at all to worry about.
I'll be using a .375 Ruger for elk this year, 270 grain Speer BTSP doing about 2700fps, I'll let you know if the bullet instantly dismembers anything.....