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.”

Louis Farrell, the director of the Student and Exchange Visitor Program, part of United States Immigration and Customs Enforcement, said academic nonimmigrant student visas require the holder “to enroll in a full course of study and attend and pass all their classes” and that “they should be wary of any recruiter who promises they can work without restrictions while attending school.”

Listed on Official Sites

But there seem to be inconsistencies in the United States Citizenship and Immigration Services’ definition of a “full course of study.” In one entry on its website, it says that the designated school officer can determine a full course of study. Farther down, it says that a student must complete at least 18 hours of classroom study.

Ultimately, the students said that because the Department of Homeland Security’s website certified the University of Northern New Jersey, they believed the institution was legitimate. In addition, the website of the New Jersey Education Department listed it as approved. So did the website of the Accrediting Commission of Career Schools and Colleges, a national body. Its director, Michale S. McComis, later said he had certified the university in order to cooperate with the government’s investigation.

Z., who is 27 and also from China, did not win the H-1B lottery. He found the University of Northern New Jersey online without a broker. “How risky will it be for obtaining full-time C.P.T. without attending any classes?” he wrote to the university in an email he showed to The New York Times, referring to Curricular Practical Training.

The response, from the generic info@unnj.edu: “If you are not comfortable with what we have offered, I ask you to try another university in the New Jersey area.”