“Do you ever hear a governor come to Washington and complain about not having enough money for their welfare program?” asked Rick Santorum of Pennsylvania, the former senator who helped devise the latest Senate health bill. “No, you don’t. Why? Because we gave them the money and the flexibility to be able to design a program that worked for them.”

But it was easier for states to do more with less when they had fewer people to worry about. After the welfare law was enacted, the number of welfare recipients plunged more than 50 percent in five years, to 5.7 million in 2001, from 12.3 million in 1996.

An equivalent drop in the number of people with health insurance or insurance subsidies would probably not be viewed as a great success.

The goal in 1996 was to help people move from welfare to work. With health insurance, the goal is to increase enrollment. And no one advocates time limits for insurance like the time limits on cash assistance.

Mr. Graham said his bill, providing $1.2 trillion for states to share from 2020 to 2026, would allow federalism to flourish. It “gets the money and power out of Washington,” he said, and “patients in this country will do better.”

But Representative Sander M. Levin, Democrat of Michigan, said the welfare program, known as Temporary Assistance for Needy Families, showed the risks of fiscal federalism.

“Those extolling the TANF block grant should check the record,” Mr. Levin said. “Federal dollars have declined by one-third in real terms, states have diverted funds for other purposes, and as a result the number of poor children served by the program has shrunk to an all-time low. If we follow this model for health care, many millions of Americans will lose their health insurance.”