Cambridge, Mass., a hub for prestigious universities and biotechnology companies, is setting aside apartments to help an unexpected group of people find lower rents: families with incomes topping $100,000.

The city recently held a lottery for 15 units with below-market rents, in a new building a few subway stops from downtown Boston. A family of four with an income well into the six-figure range—a maximum of $118,200—could qualify for some of the apartments. The city received at least 45 applications.

Many...