Her small act of independence has not come easily. To eliminate trips that could lead to falls, management at The Sequoias required residents to have walkers valet parked once they reached their table, then remain seated while staff served the meal.

That lasted until a woman sued to stop the policy, infuriated that it infringed on her freedom of movement. A compromise was reached: Residents are now allowed to take their walkers to the buffet at less busy times.

“I do like to pick out my own food,” said Mrs. Hammer, a retired music teacher. “I can tell you, as a child of the Depression, they always bring me too much.”

As the population ages and people live longer in bad shape, the number of older Americans who fall and suffer serious, even fatal, injuries is soaring. So the retirement communities, assisted living facilities and nursing homes where millions of Americans live are trying to balance safety and their residents’ desire to live as they choose.

Those who study and manage retirement facilities and nursing homes say there is heightened attention to preventing falls. Trying to anticipate hazardous conditions, retirement facilities like The Sequoias hire architects and interior designers, some of whom wear special glasses that show the building as an old person would see it.

The dangers are real. The number of people over 65 who died after a fall reached nearly 24,000 in 2012, the most recent year for which fatality numbers are available — almost double the number 10 years earlier, according to the Centers for Disease Control and Prevention.

Keeping an eye on everyone living independently is impossible, Ms. Dapper said. But a group of Sequoias staff members who refer to themselves as “the iffy committee” meet monthly to evaluate residents who seem to be growing less steady on their feet. If a cane or walker seems advisable, a staff member will suggest it to the resident, ever so gently.

Housekeepers at The Sequoias are trained not merely to clean an apartment but to inspect it for quiet hazards – scatter rugs that seem easy to trip on, or furniture that could topple if grabbed for support.

Evaluations of residents’ living quarters are regular. “We measure bed height, toilet height and the need for grab bars,” Ms. Dapper said.

In many apartments, the lip on the shower edge is black against the white tile because the contrast with the rest of the bathroom makes it more visible, said Ray Boudewyn, director of engineering for Northern California Presbyterian Homes and Services, the nonprofit company that owns and operates The Sequoias and two other facilities like it in the San Francisco area.

When finishing an apartment with trim, paint and thresholds, architects and designers put on tinted glasses to see as an older person might. The glasses cast a darkened, yellow hue. Mr. Boudewyn said he sometimes uses Vaseline to blur his glasses in order to see the rooms as patients who have cataracts would.

Carpeting is fine, and softer for absorbing a fall if it happens. But it needs to be able to accommodate an elderly person’s shuffle. As for the evenness of surfaces, Mr. Boudewyn said, “an eighth of an inch is our threshold” for tripping danger.

The Sequoias also sets standards for its residents. Everyone who applies to live there must pass a health assessment, which includes tests of cognition, gait and balance. (Depending on the type of care and apartment size, the entrance fee ranges from $142,000 to $686,000, and the monthly fee from $3,300 to $6,300.)

Yet it can be difficult to predict an individual’s risk of a fall. One of the most vigorous residents at The Sequoias, who hikes regularly and leads museum tours as a docent, recently “went down right on his face,” Ms. Dapper said.

The Sequoias has an extensive “falls education” program and an annual safety fair. But falls continue, with vexing persistence. Earlier this year, the fall rate declined slightly in both the independent-living and skilled-nursing units, according to Jill Dicus, the nurse manager at the assisted-living and memory units, but increased in the assisted-living unit.

People often resist the transition to a cane or walker, as it can feel stigmatizing. Mrs. Hammer’s own resistance ended last year after an embarrassing fall “in front of God and everybody” at her exercise class.

“When you’re using a walker, really use it and stop pretending you don’t need it,” she said. “It took the fall to destroy my confidence in walking without my walker.”

When someone does fall, residents often stay quiet about it.

Sequoias staff members encourage residents to report their falls to the nursing staff, as it indicates a risk for a subsequent fall. Yet “residents have a fear that they’ll be whisked away and put somewhere else,” said Donna Alexander, the director of nursing. She said that if a nurse observes a bruise or a cut, a fall is the prime suspect. And “the minute a resident doesn’t show up for a meal or doesn’t come to an activity, the nursing staff knows about it.”

Some residents become vigilantes. “People don’t like to say they’re turning someone in,” Mrs. Hammer said, “but I’ve heard people say, ‘You should keep an eye on so and so.’”

Yet The Sequoias can protect residents only as far as its property line – and slightly beyond, Mr. Boudewyn said. The facility spends about $30,000 a year just to maintain the sidewalks around the building. “This is a safe zone,” he said, “and the minute you leave our shell, you’re in the wild.”

One night in September 2012, while his wife was out of town, he fell in the bedroom and fractured his femur. With no way of calling for help, he lay on the floor until the next morning, when a member of the housekeeping staff found him.

This is a problem that repeats itself elsewhere. People sometimes lie in their bathrooms, doorways, kitchens or gardens for hours, even days. Still, many people resist wearing the emergency pendants. Terrence Murphy, an assistant professor at the Yale School of Medicine who studies falls, said he bought his mother a pendant to wear but she refused.

“I think most of us with aging parents go through this,” he said. “You try to reason with them, and they just don’t want to accept that they are in mortal danger if they have an injurious fall when they’re on their own. They don’t realize that deep inside their bodies, they are aging, and that when they fall alone in their house, it’s likely to be catastrophic if they don’t get help in a timely manner.”

Sometimes the adult children do not grasp the many common risks for the elderly. When Mr. Boudewyn introduced black toilet seats at The Sequoias, some of the residents’ families complained, saying they looked ugly and institutional. Then he showed them how the contrast of the black seat against the white toilet helped residents see it far better than a white seat.

Mr. Hammer’s accident changed him dramatically. He became easily agitated, and frustrated by his hampered mobility. Within a few months, he moved to the facility’s assisted-living floor and was put in a program that pinpoints residents at high risk of falling. But in a fit of aesthetic righteousness, he insisted that his furniture be placed farther apart than the staff considered safe. A few weeks later, he fell and, with nothing to grab on to, broke his wrist.

Late last spring, a neurologist gave Mr. Hammer a tentative diagnosis of Lewy body dementia, a particularly brutal form of dementia. Immediately after meeting with the Hammers, Ms. Dicus said, she started thinking about what to do next. But she wasn’t quick enough.

A week later, Mr. Hammer fell again, while walking unsupervised in the hallway of the assisted-living unit, fracturing his femur a second time. He is now in the skilled-nursing unit in a wheelchair and is not likely to leave.

A rough rule of thumb for nurses and physicians is that for every day an elderly patient spends in bed, 1 percent of muscle mass is lost. “You’d be surprised how quickly it can go below the point where you can safely stand up and move,” said Nancy K. Latham, an assistant professor at the Boston University School of Public Health who studies disability.

Had Mr. Hammer been more closely supervised, he might not have fallen, and his decline might not have accelerated. But 24-hour supervision in the assisted-living unit is unrealistic, Ms. Dicus said.

For a while, the 50-bed skilled-nursing unit of The Sequoias was like other such facilities in its use of restraints, including seatbelts for wheelchairs. But obvious physical restraints have been deemed inhumane and are associated with increased rates of complications, such as delirium, agitation and pressure sores.

Restraints have even been found to be dangerous: People belted into their wheelchairs who tried to get up would take the chair with them and both would tip over. Patients have strangled themselves on bedside rails. Hospitals continue to use restraints but are required to re-evaluate each patient’s need every 24 hours.

In the skilled-nursing unit at The Sequoias, alarms are attached to residents, including Mr. Hammer, who are at high risk for falling, triggering an alert if someone attempts to get out of bed without help.

Dr. Lipsitz of Harvard and others are not in favor of alarms like the ones used at The Sequoias. He said they often signal danger only when it is too late – after someone has already fallen down. Alarms also give rise to “alert fatigue” – they go off so often, they end up being ignored.

Hebrew SeniorLife has gradually phased out alarms at its facility in Roslindale, Mass., after finding no difference in the fall rate between those who wore alarms and those who did not. Alarms have been replaced with more frequent and more targeted rounds by nurses.

According to the Centers for Medicare and Medicaid Services, nearly 17 percent of nursing home residents in the United States, or 220,000 people, experience a fal​l.​ Of those, 70,000 are injured. Medicare currently will not pay to treat an injury resulting from a fall in a hospital, and many health policy experts believe the agency should institute the same policy for nursing homes.

“Medicare needs to focus more on preventing falls in nursing homes, including not paying to treat injuries that arise from these events,” said Dr. Ashish Jha, a professor of health policy at the Harvard School of Public Health.

In many states, if a nursing facility is found negligent, a fine is levied. Last year, for example, the state fined The Sequoias $20,000 after a resident in the skilled-nursing unit died from head trauma following a fall. In its report, the state cited The Sequoias’ failure “to provide a safe environment,” as well as improper supervision.

Asked about the fine, Ms. Dapper, the administrator, said in an email that “despite our best efforts, at times skilled nursing residents are non-compliant with their therapy precautions due to multiple diagnoses and can act in an impulsive manner that may lead to harm before anyone can reach them.”