The average employer-provided family health insurance premiums have climbed $2,976 since 2009, according to an annual Kaiser Family Foundation survey released this week. They're up $3,671 compared with the year before President Obama took office. That's despite Obama's repeated promises that the health care reform law he championed would cut premiums by $2,500 in his first term.

And while annual premium increases have moderated over the past two years, that's due to trends in the insurance market largely unrelated to ObamaCare, and trends the law could actually reverse.

The Kaiser survey found that the average family premium this year is $16,351, up 4% over last year, and up 22% since 2009. After adjusting for inflation, premiums climbed an average 3.2% a year in Obama's first term, higher than the 2.7% average during President Bush's last four years in office.

During his first campaign for president, Obama repeatedly claimed that his health reform plan would, as he said at a Virginia rally in 2008 "lower premiums by up to $2,500 for a typical family per year.

Nevertheless, the White House has been touting recent signs of health cost moderation as evidence that ObamaCare is "already working to reduce costs.

Officials cite the fact that national spending on health care climbed just 3.9% in 2011, the same as the previous two years, and the slowest increase since the 1960s.

But the trends driving the slowdown in health spending have little to do with the Affordable Care Act.

The sluggish economy played a big role. "The failure of the economy to bounce back as quickly as it has after past recessions has prolonged this dampening effect on health spending," noted Joseph Antos, a health care expert at the American Enterprise Institute.

Also important has been a broader shift that has been underway for years toward higher-deductible plans. These plans provide consumers with a stronger incentive to economize on health spending.

The Kaiser survey found, for example, that 78% of workers now face at least some deductible, up from 59% in 2008. And nearly a third has a deductible of at least $2,000, up from 12% in 2008.

In addition, the private insurance market has seen an explosion in Health Savings Account-type plans — an idea long championed by conservative Republicans — which combine high deductibles with a tax-free savings account that consumers can roll over if they don't spend it all in one year.

Kaiser found that one in five workers are now enrolled in an HSA-type plan, up from 8% in 2008.

A separate survey by America's Health Insurance Plans finds that more than 15 million people are now enrolled in HSA plans, up 15% from last year andmore than double the number in 2008.