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“I’m Anna. I’m in the history department. I want a union because I want to be able to do this work I love and have a safety net.”

“I’m Bradley. I’m from Slavic languages. I support a union because dependent health care is outrageously expensive.”

On a frigid February afternoon, 60 Columbia University graduate students from anthropology to zoology crowded into a meeting room to brainstorm how to persuade Columbia to recognize and bargain with the labor union they are struggling to form.

They began the meeting by asking the students to introduce themselves and explain why they wanted a union. At times it sounded like group therapy to air complaints — about callous faculty advisers, inadequate maternity leave and other issues — but it was far more strategy session and pep rally.

One of the last students to chime in said: “I’m Nick. I’m also in history. We need a union because power concedes nothing without a fight.”

The group said it had gotten 1,700 of Columbia’s 2,800 graduate teaching and research assistants to sign forms saying they want to unionize. More than 100 marched recently to the office of Columbia’s president to present those forms and demand that the university recognize them as a labor union. But Columbia has refused, saying that treating students as employees could hurt their education.

Columbia points to a 2004 ruling by the National Labor Relations Board that concluded that graduate teaching and research assistants were essentially students, not workers, so universities were not legally required to bargain with them even when a majority wanted a union.

Confronting that obstacle, Columbia’s graduate students are pressing the school to do what New York University did in late 2013 — agree to voluntarily recognize and bargain with its graduate students’ union.

In recent months, graduate students at Brown, Cornell, the New School and Yale have also started unionization drives, but they all face the same obstacle, the 2004 ruling. That ruling, involving graduate students at Brown, affected only private universities.

The Columbia students have asked the labor board’s New York office to oversee a unionization election, but it has refused, citing the 2004 ruling. The students hope to persuade the agency’s five-person board in Washington to reverse the 2004 ruling, which was decided by an employer-friendly, Republican-dominated board. Today’s board has a Democratic majority and is friendlier to labor.

“Given the success at N.Y.U. and some positive signals from the N.L.R.B., students at numerous schools are eager right now to pursue unionization,” said Ruth Milkman, a labor expert at the City University of New York. “There’s good reason to think the N.L.R.B. will rule their way.”

Columbia officials say the school is generous to teaching and research assistants, paying full tuition and stipends from $22,000 to $40,000, varying by department. Like many universities, Columbia fears that a union could bring tensions and strikes.

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“We fully understand that pursuing a Ph.D. is a highly challenging path, both intellectually and personally,” Columbia said in a statement. “Our graduate students are scholars in training whose teaching and research are an integral part of their doctoral studies. As the N.L.R.B. found in the Brown University case, we believe that treating students as employees could adversely affect their educational experience.”

Seth Prins, a Ph.D. candidate in epidemiology, asserted that it was wrong for Columbia and the labor board to consider the teaching and research assistants merely students. “If we stopped providing our teaching and research services to the university, it would cease to be the world-class institution it is,” he said. “Being a student is not incompatible with being a worker.”

The Columbia students are seeking to join the United Automobile Workers, which represents thousands of white-collar workers, including Columbia’s clerical workers, as well as N.Y.U. and University of Connecticut graduate students.

Alex Beecher, a fourth-year Ph.D. student in chemistry, has reservations about unionizing.

“I’m happy with the current situation,” he said. “We get a fair package from Columbia.”

“As grad students, we certainly have a worker role,” Mr. Beecher added. “But we’re also students, and by changing the nature of our worker role, a union would change the nature of our experience as students.”

At Yale, union organizers have collected more than 1,000 pro-union signatures from graduate students and asked the university to grant recognition. More than 1,000 graduate students and their allies staged a protest on campus in October, with Gov. Dannel P. Malloy of Connecticut there to voice support.

“This has really been energized by the example of N.Y.U.,” said Aaron Greenberg, a graduate student in political science and president of Yale’s Graduate Employees and Students Organization. “We want to sit down and negotiate about our benefits, our pay and our work at the university.”

A university spokesman, Thomas Conroy, said: “Yale believes that graduate students are students, not employees. Further, we do not believe it is in the best interests of the students, the faculty or the educational process to change the teacher-student relationship to a manager-employee relationship.”

Mr. Conroy called Yale’s financial package for graduate students “among the most generous in the country,” valuing it at $65,000 a year. The students are guaranteed five or six years of full tuition plus stipend and heavily subsidized health coverage.

Luis Lei, a graduate student in philosophy at the New School, spoke in envy of Columbia’s and Yale’s financial packages. At the New School, graduate students generally pay most of their tuition and receive modest stipends for teaching or research.

A majority of the New School’s graduate students have signed forms supporting membership in the autoworkers’ union.

“The basic concerns are wages, a lack of job security and not having any say about labor conditions,” Mr. Lei said.

The New School has not responded to the students’ request for union recognition, saying it is evaluating its options.

“We are assessing the impact recognition would have on the education of all of our students,” a New School spokeswoman, Josephine Parr, said. “Our current resources are more modest than those of larger research universities.”

At Columbia, Olga Brudastova, a doctoral student in mechanical engineering, voiced a common complaint: She said her stipend checks sometimes arrived three months late. When that happens, graduate students worry that if they push too hard to get their delayed checks, they might infuriate their faculty adviser or the administration.

“Having a union could go a long way to help,” Ms. Brudastova said.