This is a fascinating question that goes to the very heart of this excellent and moving short story about a young man who prefers the world of illusion to the world of reality. Certainly, overtly Paul's decision to kill himself rather than return to his middle-class roots would suggest that he is a tragic figure. His fatal flaw in his character clearly is his love of art and beauty which leads him to commit an act of crime to live his dream existence for just a short time before the game is up.
However, crucially, Cather leaves it very open as to the reason for Paul's downfall, and leaves us as the readers to decide. Is he an attractive dreamer, a romantic born into the gritty middle class realism that he detests, or is he just simply an impractical character that needs to grow up and face reality? Is he a "bad case" as others decide, or does he have some redeeming features and are they too narrow in their terms of definition? To what extend is Paul responsible for his own downfall, and to what extent is it American materialism and vice that destroys him?
No definite answers to these questions are presented, but the story demands that each one of us makes up our own mind. Although partly I do find Paul an intensely annoying character whom I think needs a good shake, there is also something attractive in his desire to pursue dreams and illusions to their bitter end. Although I disagree with his actions, I appreciate his overwhelming desire to act on his dreams and let them take him to New York and to the logical, bitter end of his life. In this sense, I feel he is a tragic character, as he is always haunted by his limitations and what he has not achieved. Note his final thoughts as he jumps to hit the oncoming train:
As he fell, the folly of his haste occurred to him with merciless clearness, the vastness of what he had left undone. There flashed through his brain, clearer than ever before, the blue of Adriatic water, the yellow of Algerian sands.
Paul is a character whose dreams are to large to be contained in this life, and thus perhaps it is fitting that at the end of the tale he drops back "into the immense design of things" from which he emerged.