Despite President Obama’s assurances to the contrary, millions of Americans are discovering that they will not be able to keep their current health insurance when the Affordable Care Act takes full effect next year.

Reports Tuesday from CBS and NBC News document this development. According to CBS, US insurance firms have sent health policy cancellation notices to more than 2 million people, with 800,000 in New Jersey alone.

NBC estimates that 50 to 75 percent of the 14 million Americans who purchase their health insurance as individuals on the open market can expect a cancellation letter, as their policies don’t meet standards mandated by the ACA.

Mr. Obama has long insisted that under his signature domestic achievement Americans who like their current health plan can keep it. That’s turning out to not be exactly true as the changes produced by the sweeping Obamacare law work their way through the US health system.

But the White House is pushing back against the notion that Obama was misleading, saying that the cancellation notices are in part the result of normal turnover in the individual insurance market.

“What the president said and what everybody said all along is that there are going to be changes brought about by the Affordable Care Act to create minimum standards of coverage,” said White House Press Secretary Jay Carney.

So what’s going on here? It’s a complicated subject, so we’ll try to explain it as simply as we can.

First, what’s at issue here are health insurance policies that people buy themselves on the open market, because their employers do not offer insurance, they work for themselves, or they don’t work and don’t qualify for Medicare or Medicaid. It is not about employer-provided health plans, which is how most workers get their coverage.

Second, Obamacare regulations raise the bar for such plans. The ACA requires that they provide fuller and more complete coverage than they did before. Many old cheaper plans that offer coverage only for catastrophic events or that have high deductibles may not meet these new rules.

Third, there is a provision in the ACA that grandfathers in plans in existence at time of the law’s passage in 2010. But there’s a catch: They had to remain unchanged to qualify for this status. Any alteration in beneficiaries, co-pays, coverage, or other relatively routine item meant they had become new plans in the eyes of Obamacare.

So that’s the context. What’s happening now is that people with un-grandfathered individual plans that don’t meet new rules are getting notices that their current coverage is being canceled. In most instances, they’re being offered a new policy that’s Obamacare-compliant. But in some, maybe many, cases, that new policy is more expensive, because it offers more.

Middle- and lower-income Americans might qualify for government Obamacare subsidies to pay for this coverage, of course. But they’re still facing sticker shock and a loss of their current insurance arrangements.

“An accurate way to portray what’s happening might go like this: Millions of people are finding out they’re being rolled over into new policies that provide fuller coverage. But some may end up paying more, even after the subsidies that the law provides,” writes left-leaning TalkingPointsMemo editor Josh Marshall in a defense of Obamacare policies.

The political problem for the administration is that Obama’s assurances that “if you like your doctor or health plan, you can keep it” have turned out to be an oversimplification. Republicans have seized on the issue as yet another example of government overreach. They note that many people were satisfied with cheaper individual policies that are no longer available due to the ACA. That may be particularly true for younger, healthier people who are now facing premium increases to offset the costs of older and sicker enrollees.

“Proposed new federal law: Everyone has to buy their cable network’s most expensive package. More channels for you!” tweeted the right-leaning commentator Allahpundit on Tuesday.

Was Obama’s promise a falsehood? Current New Jersey governor and possible future GOP presidential candidate Chris Christie on CBS “This Morning” Tuesday said flatly that “people weren’t told the truth.”

In a more nuanced view, the generally left-leaning Jonathan Chait writes in New York Magazine’s “Daily Intelligencer” that there are larger goals in the new rules, such as a reduction in the number of medical bankruptcies and improvement of generally skimpy low-cost coverage.

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Years prior to implementation, the notion of imposing new regulations on the individual health-care market did not feel as if it were taking people’s coverage away from them.

“In the current moment, with cancellation notices going out and alternatives not yet available, it feels exactly like that,” Mr. Chait writes. “Which is to say, a promise that felt like a mere oversimplification at the time, and may eventually feel like one in retrospect, currently feels like a lie.”