Like many schools across the United States, St. Mary’s, a small, public liberal arts college, is figuring out how to field increasing requests for animals by students with diagnosed mental health problems. Last fall it began allowing “comfort animals” for students like Ms. Brill, Theo’s owner, who has anxiety and depression, and Ms. McCarthy, Carl’s owner, who gets panic attacks.

Anxiety, followed closely by depression, has become a growing diagnosis among college students in the last few years. The calming effect of some domesticated animals has become so widely accepted that many schools bring in trained therapy dogs to play with stressed students during exam periods.

But as students with psychiatric diagnoses are asking to reside on campus with their own animals, schools with no-pet housing policies are scrambling to address a surfeit of new problems. How can administrators discern a troubled adolescent’s legitimate request from that of a homesick student who would really, really like a kitten? If a student with a psychological disability has the right to live with an animal, how should schools protect other students whose allergies orphobias may be triggered by that animal?

The topic is being hotly debated by college housing and disability officials in the wake of discrimination lawsuits filed by students who were denied so-called emotional support animals. Last month, on the eve of a trial in a case closely watched by administrators, the University of Nebraska at Kearney settled with the Justice Department, agreeing to pay $140,000 to two students who had been denied support animals, and spelling out protocols for future requests. Recently, a federal judge refused to dismiss a similar case against Kent State University.

“The disabilities services people are all looking at what they need to do to make this work,” said Jane Jarrow, an educational disabilities consultant who is teaching “Who Let the Dogs In?” — an online course about emotional support animals — for the fourth time this year. “We’re way past pretending it’s not going to happen.”

In the years before support animal lawsuits, universities found it relatively easy to say no to requests for animals. But now, said Michael R. Masinter, an expert on disability law at Nova Southeastern University in Florida, “schools think it’s easier to say yes than no because property damage is cheaper than litigation.”

Perhaps that explains the 95-pound pig that a freshman was allowed to bring to her second-floor room at Washington State University. Unfortunately, when led to the stairs, the pig balked. The freight elevator made him anxious, too. So he stayed in the dorm room and used a litter box.

“The other students thought the pig was kind of cool, but less cool when it began to smell,” recalled Hannah Mitchell, the dorm’s residential director at the time. “We talked about bathing it. But dorm bathrooms aren’t built for washing animals.”

Pig and student transferred to a dorm with ramps. Eventually, both moved off campus. Custodians said the dorm room’s carpeting had been chewed-up, the furniture gnawed and closet doors knocked off.

The overwhelming majority of support animal requests are for dogs and cats. But schools have had requests for lizards, tarantulas, potbellied pigs, ferrets, rats, guinea pigs and sugar gliders — nocturnal, flying, six-ounce Australian marsupials.

Clearly, many requested animals are students’ pets. But what is the difference between an emotional support animal and a pet that also provides support?

The distinction depends less on the animal and more on the student: whether the student has a diagnosed psychiatric disorder, and can document that the animal is therapeutically necessary.

“Do we have people trying to get their pet across as an assistance animal? Sure,” said Jamie Axelrod, director of disability resources at Northern Arizona University, where requests for support animals rose to about 75 last year from a handful a few years ago. “Do we have people who legitimately require one? We do.”

“You have to rely on a treatment provider’s ethical sense that they’re doing what’s right for their patient,” Mr. Axelrod added. “But it’s a new gray area.”

Research on the therapeutic value of animals is limited. Some studieshave shown that they can provide a short-term benefit, particularly in reducing anxiety and depression. A long-term therapeutic benefit, however, has not been definitively established by randomized control trials.

Joanne Goldwater, associate dean of students and director of residence life at St. Mary’s College of Maryland, is not concerned about objective evidence. “Having that animal has clearly helped to reduce stress and anxiety for some students,” she said, “which helps them progress towards their degree.”

Students concur. Ms. Brill, a film major, wrapped her arms around Theo’s neck. “Theo helps me when I’m feeling isolated and depressed,” she said. On wobbly days, he gives her structure, she added, because she must get out of bed to feed, brush and walk him. “All I have to do is look at Theo, squish his face a lot in the evenings, and he’s like, ‘Hey, I love you!’ ”

Her roommate Ms. McCarthy, a psychology major, tucked Carl into her neck, stroking his silky fur as he eagerly nuzzled her ear. “When I feel a panic attack coming on, feeling his heartbeat helps me regulate my own,” she said.

And animals have inspired creative compromises. At St. Mary’s, animal owners must do their laundry in designated washing machines and dryers to avoid cross-contamination with the clothing of students with animal allergies.

At Western Washington University, a student asked to keep her six-foot snake. But the school prohibits “live feeds,” said Karen M. Walker, the associate housing director. The solution? Frozen mice, served thawed — a solution amenable, so far, to the school, student, suite mates and snake.

Whether schools must permit support animals depends, generally, on federal housing law. The Nebraska suit was filed by the Justice Department in 2011 on behalf of Brittany Hamilton, whose four-pound miniature pinscher, Butch, would put his paws on her shoulders to quell her anxiety attacks. She wanted Butch to live in her university apartment. The university said no.

But in 2013, a federal judge ruled that the university’s residences were bound by the Fair Housing Act, which protects individuals with disabilities from discrimination. Among the act’s “reasonable accommodations” for residents with psychological disabilities are animals that provide emotional support.

While the consent order in the Nebraska case last month is not binding on other colleges, it lays out some guidelines. The university can deny a request if the animal is too big for the quarters or aggressive, or damages property.

And if a student’s documentation looks insufficient, a school can contact the student’s medical provider — a pushback against emotional support animal letters downloaded on the Internet or churned out by cybertherapists who, for fees of up to $150, will Skype with the student and then issue the document. Universities have been circulating a watch list of such practitioners.

Some institutions are managing the issue with a matter-of-fact attitude. “We use our code of conduct for animals as well as people,” said L. Scott Lissner, the Americans With Disabilities Act coordinator at Ohio State University. “We don’t let our students walk across campus and lick people unless it’s welcome, so we don’t let the dogs do it. We don’t let students howl all night.” And, he added, “they can’t go to the bathroom wherever they want.”