WASHINGTON — As health care legislation moves toward a crucial airing in the Senate, the White House is facing a growing revolt from some Democrats and analysts who say the bills Congress is considering do not fulfill President Obama’s promise to slow the runaway rise in health care spending.

Can the United States afford to revamp its health care system?

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Mr. Obama has made cost containment a centerpiece of his health reform agenda, and in May he stood up at the White House with industry groups who pledged voluntary efforts to trim the growth of health care spending by 1.5 percent, or $2 trillion, over the next decade.

But health...