If O. Henry's short story "After Twenty Years" had not been told in the third person, it would have to have been narrated in the first person either by Jimmy or by Bob. Obviously, allowing Jimmy to tell the story would have destroyed the suspense and the final twist completely. Also, he would have to have hidden somewhere or performed some other sleight of hand to explain his being able to give an account of the conversation between Bob and the plain-clothes police officer. To have given the story to Bob, on the other hand, might have preserved the mystery, but there would have been too little variation in the tone, since Bob speaks in very simple, declarative sentences and is not much given to description.
In fact, the reader has the best of both worlds—since the dialogue which forms most of the story contains a first-person narrative by Bob and the story ends with a note from Jimmy, neither of which would have been so effective without the framing of a third-person narrative.