As with the Smollett case, the sensational elements of the story — a young Muslim woman cornered, harassed and followed — became fodder for the news media.

“These kinds of rare cases do damage, especially in the current hostile political environment,” said Ibrahim Hooper, a spokesman for the Council on American-Islamic Relations. “They are seized upon and used to try to discredit legitimate anti-Muslim incidents or used to say there is no such thing as a hate crime.”

Mr. Hooper said his group received 2,000 to 3,000 bias reports annually before verification, and “less than a handful” were deemed false each year.

Cynthia Deitle spent more than 20 years as an F.B.I. special agent in the civil rights unit, focused on hate crimes and police brutality. As the director of programs and operations at the Matthew Shepard Foundation, she said the most common motivation behind the false hate crime reports that she had investigated was the desire for attention and sympathy.

“They felt that being labeled a ‘hate crime victim’ would be a positive result,” Ms. Deitle said. “The cases I examined took place in a workplace environment and at residences, and sometimes the allegation was made to draw attention away from a real negative aspect of the complainant’s life, like poor performance at work or school, or a feeling that the complainant was not getting the attention she or he deserved.”

In Mr. Smollett’s telling of the Chicago attack, he was leaving a Subway restaurant in the Streeterville neighborhood when two masked men attacked him around 2 a.m. He said they called him homophobic and racial slurs and slung a rope around his neck. Even by hate crime standards, the story of an assault on a celebrity seemed shocking, and garnered relentless attention.

But it wasn’t long before the story collapsed. Questions, inconsistencies and a lack of evidence chipped away at Mr. Smollett’s version of events. He steadfastly pushed back at doubters, even appearing on “Good Morning America,” where he passionately proclaimed his innocence and questioned why anyone would make up a hate crime. “You do such a disservice when you lie about things like this,” Mr. Smollett told Robin Roberts.