As many as one in four students at some elite U.S. colleges are now classified as disabled, largely because of mental-health issues such as depression or anxiety, entitling them to a widening array of special accommodations like longer time to take exams.

Under federal law, students can be considered disabled if they have a note from a doctor. That label requires schools to offer accommodations depending on the student’s needs. A blind student, for example, would have access to specialized software or a reader for an exam.

The rise in disability notes for mental-health issues has led to a surge in the number of students who take their exams in low-distraction testing centers, are allowed to get up and walk around during class or bring a comfort animal to school, among other measures.

“At Pomona, we have extremely talented bright students with very high expectations who are coming in with a good level of anxiety and are highly stressed,” says Jan Collins-Eaglin, the Claremont, Calif., college’s associate dean of students for personal success and wellness. “Our job here is to help them really thrive.”

At Pomona, 22% of students were considered disabled this year, up from 5% in 2014. Other elite schools have also seen a startling jump in disabilities, according to data from the federal government and from the schools. At Hampshire, Amherst and Smith colleges in Massachusetts and Yeshiva University in New York, one in five students are classified as disabled. At Oberlin College in Ohio, it is one in four. At Marlboro College in Vermont, it is one in three.

Small, private schools have the greatest concentration of students with disabilities. Among the 100 four-year, not-for-profit colleges with the highest percentage of disabled students, 93 are private, according to a WSJ analysis of federal data.

The fraction of college students with genunine disabilities cannot have quadrupled in just a few years.