The proof of Godel’s theorem is based on the existence of a program which correctly checks proofs. So I would say that proof checking is a classic example, perhaps the classic example, of something that can be done correctly. In fact, that is literally the assumption of Godel’s theorem—it says that if proofs in an axiom system can be checked using a computer, then there are statements in the language which can’t be proved.
The proof of Godel’s theorem is based on the existence of a program which correctly checks proofs. So I would say that proof checking is a classic example, perhaps the classic example, of something that can be done correctly. In fact, that is literally the assumption of Godel’s theorem—it says that if proofs in an axiom system can be checked using a computer, then there are statements in the language which can’t be proved.