I had a friend who was practicing with his new .416 for his upcoming Cape Buffalo hunt. He was using it to shoot out of his second story window at ground squirrels in his pasture. Any squirrels that he missed got new holes to hide in.
To specifically answer your question, it is my understanding that the .22-250 was originally developed as a varmint cartridge, and with the right bullets it has been successfully used on deer sized (and larger) animals. I've shot both mule deer and antelope with mine. I have friends who have shot elk and buffalo with theirs.
And the .25-05 was originally developed as a deer cartridge that would also work well on long range shots on varmints on windy days. I've used my .257 AI, which is balistically similar to the .25-06, on everything from prairie dogs to elk. It's my favorite deer, antelope, and bighorn sheep rifle.
The record books, that are put out by Boone & Crocket and Pope & Young and where I live is Henry Kelsey they are a great way to research a state or a province for big game. This does take some time but the knowledge it gives you, can be very valuable. Not all hunters have their trophies recorded, as where I live after the Hanson Buck became the new world record typical Whitetail, we saw a flood of hunters and now that has happened, some hunters are shying away from the record books but...