Any more genres and we'd be in kitchen-sink territory. The thing is, all these genres are mixed up together, so you really can't have one without the other. Let's start with "Family Drama":
Relations between Paul and his family are strained, to say the least. Paul is gone as much as possible, sneaking in late to avoid "his father at the top of the stairs, his hairy legs sticking out from his night-shirt" (1.20). Ask Paul a question? He's going to lie about it. Not to mention that he doesn't even seem to have a relationship with his sisters, and, as in all good family dramas, his mother died when Paul was a baby.
So, basically, not a warm and nurturing environment for this fragile flower.
Crummy family life is just one step away from Gothic, as anyone who's ever read Edgar Allan Poe's "The Fall of the House of Usher" or Stephen King's The Shining can tell you. Gothic stories often contrast beautiful, fantastical places (inside Paul's hotel room, inside the Waldorf dining room) with dark, scary places (Paul's bedroom, Paul's basement). These contrasts help emphasize the troubled, depressed state that Gothic characters always seem to find themselves in. (Well, no wonder.)
And Gothic interest in the troubled minds of their characters is just one step away from the psychological thriller. "Paul's Case" builds tension and suspense as Paul spirals further and further downward into depression. Why exactly Paul is so depressed? What he's doing that he's so ashamed of? What are his horrible fears? And what's going to happen at the end?
Okay, so how can a story be Gothic and realist at the same time? Well, Cather uses real places—Carnegie Hall, the Waldorf Hotel—as her settings, and she describes these places in accurate detail. She focuses on the mundane details of life, like hairy legs and worn-out clothing and "boots [that] were letting in the water" (1). Not to mention that her portrait of a suicidal student is written almost as though it's an entry in a clinician's case file.
That brings us to our last genre: tragedy. Our hero (such as he is) dies at the end of the story, which is the basic definition of tragedy. What's worse is that Paul regrets his decision to commit suicide at the exact moment he's throwing himself in front of the train. Totally, totally tragic.