“Paul’s Case” is a fictional short story. Although it is not a play and thus wouldn't normally be classified as a tragedy, it does follow the traditional narrative structure of a tragedy, insofar as it ends with Paul's downfall. More specifically, though, the story plays with certain features of Realism, considering that it's primarily interested in exploring Paul's desire to break free from an uninspiring quotidian life. By featuring a dreamer who effectively rejects everyday mundanity in favor of a more imaginative and artistic way of moving through the world, the story ultimately pushes against the conventions of Realism, a genre that sought to portray normal life in a straightforward manner.
Still, this doesn't change the fact that the story has a tragic end. From the very start of the short story, Paul's case is hopeless. The fault is not with him, but with the social roles and expectations that keep him tethered to the upbringing he despises. After the climax, there remains no pretense that Paul's case can resolve happily; he simply wishes to burn brightly before his ultimate destruction. Thus, his eight days of debauchery in New York and his ultimate suicide resolve Cather's realist and modernist tragedy.