“It is very much like a honey trap because you have a very, very great temptation being waved in front of these young men and women,” said David Grunblatt, co-head of the immigration and nationality group in the labor and employment law department at Proskauer Rose in New Jersey. He represents companies concerned about hiring foreign graduates for practical training.

Most of the students were from China and India, where working with brokers is a familiar way of doing business. But in retrospect, they seem to have ignored what should have been red flags, whether because they were overly trusting, willfully ignorant or willing participants in visa fraud.

Corey Lee, an immigration lawyer in Manhattan, said: “If you didn’t go to a class for a year, you should expect something is going on.”

Ms. Sun, 29, earned a master’s degree in art therapy at a Massachusetts college and moved to New York to work for a nonprofit social services agency. (She declined to name the college or the organization because she did not want to endanger her application for a waiver to return to the United States.)

Her employer applied for an H-1B visa for her, but she faced a gap of several months before it would come through.

Ms. Sun said her broker told her she could work instead of taking classes, but referred her to the university. In February 2015, after she enrolled, Ms. Sun said she called there approximately 20 times; no one responded.

“If one person called me back and told me it was fake, I would not have gone to the University of Northern New Jersey,” Ms. Sun said. “What else can I do? I don’t know the American system.”