To what extent is the protagonist of the story "Paul's Case" responsible for the conflict or predicament he or she faces?

The more superficial answer is that Paul is entirely responsible for the predicament he finds himself in. He makes the audacious decision to steal a large amount of money from his employer and go on an upper-class spending spree, living the high life for a short time in New York City. He is then faced with the choice between arrest and imprisonment and committing suicide. He chooses suicide, a fate he brought on himself.
But looking a little deeper, we might realize that Cather also implicates the values of the society Paul lives in and has internalized. If you live in a society that values wealth and luxurious living, is not the society itself also at fault if you pick up those values and want those things that society tells you are the most important sources of self worth? Paul is willing to price the price of those false values, which is to trade his life for a few luxurious days.
Paul's problem is that he places far too much value on the shallow surface of life, such as what clothes a person wears or where they dine out. However, he didn't spring from the womb holding those values: he learned them from the larger culture. Paul's predicament is his society's predicament of placing too much importance on the external and shallowly material rather than on internal worth.

One could argue that Paul is entirely responsible for his predicament. Throughout his life, he's consistently made the wrong choices. Instead of facing up to his responsibilities, he's locked himself away in a fairy-tale world where he gets to act out his fantasies of being a rich member of elite society. Yet despite his unwavering admiration for the rich, Paul does absolutely nothing to improve his station in life. He's not prepared to knuckle down and do the hard work necessary to take him where he wants to be. Instead, he takes a shortcut by stealing from his employer. But even then, he only uses his ill-gotten gains to live out the fantasy of being a wealthy young man-about-town.
Paul has demonstrated time and time again that he's incapable of living in the real world. So it comes as no surprise when, in the story's tragic ending, he takes his own life. Ironically, this is the first—and last—time that Paul has ever taken responsibility for anything. And it's the first—and again last—time that he's been able to resolve the conflict between the real world and the fantasy world in which he's lived for so long.