WASHINGTON (AP) — President Barack Obama’s soothing promise that Americans happy with their health insurance could simply keep it was doomed from the start, and everyone familiar with the market seemed to recognize that except the president.

But he kept making the promise, through the long debate about the overhaul, after it became law and directly to voters in the campaign for the 2012 election. Now his assurance is proving empty for people who are getting cancellation notices in the individual or small-business insurance marketplace and for workers who are beginning to see jarring changes in their employer-provided plans. Although they are a minority of the insured, they are adding up to millions of people.