As a matter of fact, the late Jack O'Conner's wife killed her elephant with a 30-06 shooting a 220gr. solid.
The great elephant hunter Karmajo Bell killed over 1000 elephants with the 7x57 mauser. So yeah it can be and has been done many times over. Again bullet placement is the key. A deer will drop quicker from a well placed shot from a 223 than a bad shot from a 375 H&H or any other caliber. Again, BULLET PLACEMENT!!!!