Why is Niles So Broke?

If he was working steadily as a psychiatrist, he probably made very good money, but if he may have made the mistake of pooling resources with his wealthy wife. In a divorce, the party who acts to freeze mutual assets first has a huge advantage, as does the partner who can afford the best lawyers, and the partner who is the most ruthless. If Maris got the drop on him in all three ways, then during the separation, he'd be stuck with his own current income and massive legal bills.

So yes, it's believable that a working person who is divorcing someone rich and ruthless would go through a period of severely reduced expenses, although of course he'd be well off enough in the long run. There would be a long-delayed settlement from the divorce and his own income, and he'd be well off enough even if his lifestyle was never what it had been with Maris. From then on, he'd be enough of a twat to call himself "Poor but happy".

Niles was deeply in debt from lavishing money on male prostitutes. And he was being blackmailed by one of them.