More people are experiencing mental-health problems at work—stresses that often worsen around the holidays. While employees with mental illness have traditionally kept quiet about it for fear of being stigmatized, some are finding new ways to discuss their needs with bosses and colleagues.

Still, some workplaces remain hostile to the whole idea. Chris Jackson of Los Angeles suffered anxiety, stress-induced headaches and nighttime panic attacks on a high-pressure sales job at a previous employer. But he knew asking for a mental-health day would read as weak in the firm’s macho, competitive culture.

“I told my manager I suffered a few concussions in high school football, which I did not play, to pander to the boys-club mentality. I said my brain had been rattled, causing migraines and light sensitivity,” Mr. Jackson says. At 6-foot-4 and 235 pounds, he could pull off the fib, and was able to take a day or two off now and then without damaging his reputation. “I was seen as a warrior jock, rather than a weak link,” says Mr. Jackson, a financial planner and CEO of Lionshare Partners, an investment advisory firm.

Like any other sick time, a mental-health day can cover a range of problems, from simple fatigue or burnout to more serious problems like chronic depression or an anxiety disorder.

Some employers have restrictive sick-day policies that cause employees to hide mental-health issues. Matt Garrett, an Atlanta photographer and digital-marketing specialist who has bipolar disorder, once worked at a company that required a doctor’s note before granting a sick day. When he needed time off to manage his symptoms, he pretended to have a cold. “The doctor would send me home with a note to get ibuprofen,” Mr. Garrett says.

Employees with a mental illness are generally protected from discrimination by U.S. disability-rights law, but few care to make a federal case of it. “It’s easier to just go somewhere else,” Mr. Garrett says. He later negotiated the right at a different employer to take an unpaid day off for any reason, no questions asked.

Some 18.9% of Americans report having mental illness in the past year, up from 17.7% a decade ago, federal data show.

Depression and other mental-health problems are the top reason employees call for help from ComPsych, a Chicago-based employee-assistance provider, according to a recent analysis of two million calls. Stress and anxiety recently overtook relationship problems as the No. 2 issue, outpacing such issues as child-behavior problems and drug or alcohol addiction.

Employees in their 20s report higher rates of stress, anxiety and depression than other age groups, ComPsych says. And younger workers are more likely to speak openly about it than older ones.

Marketing strategist Andrew Clark of Towson, Md., finds it helpful to talk with his boss about how anxiety and depression affect his work.PHOTO: NOAH KAIN

Andrew Clark, who suffers from anxiety and depression, needed a mental-health day last summer just a few weeks after starting a new job as a marketing strategist at Duckpin, a digital-marketing agency in Towson, Md. The 31-year-old texted his boss, Noah Kain: “Hey, Noah, I’m having some mental-health issues and I feel I need to address them immediately.”

Mr. Kain, also 31, had never received a request for a mental-health day. He’d seen co-workers on previous jobs abuse time-off privileges, but he decided to trust Mr. Clark. “If he had mental-health issues, I wanted to be helping him with that, rather than causing him any problems,” Mr. Kain says. He texted a sympathetic response: “Hey, no worries, I’ve been there,” and followed up later that day to make sure Mr. Clark was OK.

Employees who need time off because of a diagnosed mental illness are typically protected from discrimination by the Americans with Disabilities Act. Also, the federal Family and Medical Leave Act may provide job-protected leave without pay, says Frank Alvarez, leader of the disability-leave practice in White Plains, N.Y., for the law firm Jackson Lewis.

U.S. law doesn’t protect abusers of these policies. “Some employees use this phrase mental-health day to cover up for the fact that they just want a day off because they have something better to do. Maybe they received a last-minute free ticket to ‘Hamilton,’ ” Mr. Alvarez says.

Such tensions pose a challenge to managers. “You have to weed out the bad seeds from the employees who have legitimate issues,” says Gillian Florentine, a Pittsburgh human-resources consultant. Employers also need to know if job demands are so unreasonable that they’re causing mental-health problems. Bosses still need to respect employees’ right to privacy, however, and avoid probing for sensitive information about individuals’ mental health.

More employers are freeing managers from the need to decide whether employees’ excuses are acceptable by replacing sick-day policies with paid personal days employees can use for any reason. Some 36% of employers offer this kind of time off, up from 22% in 2014, according to a 2018 survey of 3,518 employers by the Society for Human Resource Management.

Esther Gonzalez Freeman, a speaker and business coach, sometimes took mental-health days on a previous job as a university administrator. PHOTO: ESTHER GONZALES FREEMAN

Many employees say it’s still important to speak up about what is increasingly a hot-button topic. When Ben Congleton praised one of his employees last year for asking openly for a mental-health day, defying the stigma, the employee’s screenshot of the email exchange was retweeted more than 16,000 times and drew 44,000 likes. Mr. Congleton is CEO of Olark, a live-chat software provider in East Palo Alto, Calif.

In a follow-up essay in Harvard Business Review in January, Rhoda Meek, Olark’s customer-service director, described the benefits of talking openly with her team about her challenges with bipolar disorder.

Esther Gonzalez Freeman says it took years of work experience as a university administrator to set aside her shame about having depression and anxiety and talk with her supervisor about it. The conversations brought them closer, says Ms. Freeman, a Marietta, Ga., business coach and speaker. “I have never met anyone who was ashamed of their allergies or diabetes. So why should mental health be treated any differently?” she says.

“We talk about mental health in a reactive way, after a Kate Spade or an Anthony Bourdaincommits suicide. Part of me wonders, if we were to normalize talking about mental health, how many people could we keep from that kind of suffering?”

Need a Mental-Health Day?

• Know your company’s time-off policies before asking.

• Be aware that some managers hold damaging stereotypes about employees with mental illness.

• Watch how your bosses handle colleagues’ time-off requests and adapt yours accordingly.

• Try if possible to be honest rather than pretending your ailments are physical.

• Share only as much information as bosses need to know.

• Try to manage your work so colleagues aren’t swamped because you’re out.

• Avoid using a mental-health day as a pretext for playing hooky.

Write to Sue Shellenbarger at sue.shellenbarger@wsj.com