Democratic presidential front-runners Bernie Sanders and Elizabeth Warren are confronting the same challenges to their health-care policies that foiled state lawmakers who tried to launch universal health systems in recent years.

Legislators in Vermont, Mr. Sanders’s state, backed Medicare for All before figuring out a detailed financing plan, and the final numbers were too high to sell politically. Colorado weighed its own version of guaranteed health care, but public support fell as the industry ramped up an opposition effort....