Earlier this semester, college senior Leah Nelson emailed one of her instructors to ask for extra time to complete a paper. "I have been going through a rough patch lately and am making the decision to take care of myself this week," Ms. Nelson wrote. Her mental health, she continued, would "take priority over everything else."

Ms. Nelson, a 21-year-old student at the University of Connecticut in Storrs, struggles with depression. Her symptoms often flare when exams and papers pile up. She says the timing of a suicide attempt...