HOUSTON — Wealth has never had a stigma in the affluent suburbs of Fort Worth, where the town of Westlake landed, to no one’s surprise, on Forbes’ list of America’s most affluent neighborhoods last year with a median income of $250,000.

But in recent days, the implications of being rich have set off an emotional, angry debate that has stretched far beyond the North Texas suburbs, after a juvenile court judge sentenced a 16-year-old from a well-off family to 10 years’ probation for killing four people in a drunken-driving crash.

The judge, Jean Boyd, on Tuesday declined to give the teenager, Ethan Couch, the punishment sought by Tarrant County prosecutors — 20 years in prison — and instead ordered him to be placed in a long-term treatment facility while on probation. Judge Boyd did not discuss her reasoning for her order, but it came after a psychologist called by the defense argued that Mr. Couch should not be sent to prison because he suffered from “affluenza” — a term that dates at least to the 1980s to describe the psychological problems that can afflict children of privilege.

Prosecutors said they had never heard of a case where the defense tried to blame a young man’s conduct on the parents’ wealth. And the use of the term and the judge’s sentence have outraged the families of those Mr. Couch killed and injured, as well as victim rights advocates who questioned whether a teenager from a low-income family would have received as lenient a penalty.