On a Sunday evening a few weeks back, Shobana Ram was loading the dishwasher in her kitchen in Queens when her 85-year-old father-in-law rose from the dinner table, carrying his cane in one hand and an empty plate in the other.

“From the corner of my eye, I saw him stumble and lose his balance,” recalled Ms. Ram. “I saw the cane fly out of his hand. His head hit the corner of our granite countertop.”

She dialed 911 and thought, not for the first time, how fortunate it was that in 2016 she and her husband sold their house and bought one big enough to accommodate six people: themselves, their two teenagers and his ailing parents, plus the family dog.

Her mother-in-law, who has dementia, would not have been able to phone for help. In this case, after emergency room scans, her father-in-law was “miraculously OK,” said Ms. Ram, 48.