President Barack Obama's signature on the health insurance reform bill at the White House, March 23, 2010. President Obama promised the legislation would help reduce health insurance premiums, but growth has continued. (Official White House Photo by Chuck Kennedy)

Two new independent studies of health insurance premiums and health care spending indicate both are rising at an accelerated pace, despite President Obama's 2008 promise to contain those costs and his pledge that his health care legislation would reduce premiums.

Spending on health care rose 4.6 percent in 2011 - up $4,500 per person, on average - according to the nonpartisan Health Care Cost Institute. That's up from a 3.8 growth rate in 2010.

Health insurance premiums for individuals and families also climbed year-over-year, up 3 percent ($186) on average for an individual and 4 percent ($672) on average for a family, according to the Kaiser Family Foundation.

During Obama's term, between 2009 to 2012, premiums have climbed $2,370 for the average family with an employer-provided plan - a rate faster than the during the previous four years under President George W. Bush, according to Kaiser.

Investor's Business Daily's John Merline was first to note the difference in premiums climbing faster under Obama than the previous four years under Bush.

Experts point to rising health care costs as the driver of increased individual spending and higher premiums.

During the 2008 campaign and health care reform debate in 2009, President Obama said repeatedly that his plan would bend the cost curve downward, ultimately saving the average family $2,500 per year.

At a rally in Virginia in June 2008, Obama said: "In an Obama administration, we'll lower premiums by up to $2,500 for a typical family per year."

"We'll do it by investing in disease prevention, not just disease management; by investing in a paperless health care system to reduce administrative costs; and by covering every single American and making sure that they can take their health care with them if they lose their job," he said at the time. "We'll also reduce costs for business and their workers by picking up the tab for some of the most expensive illnesses.

"We won't do all this twenty years from now, or ten years from now," he said. "We'll do it by the end of my first term as President of the United States."

And in a debate with Sen. John McCain in October 2008: "The only thing we're going to try to do is lower costs so that those cost savings are passed onto you. And we estimate we can cut the average family's premium by about $2,500 per year," he said.

Obama and Democrats enacted the Affordable Care Act, a.k.a. "Obamacare," in March 2010, including many of the projected cost-saving measures that were hailed before voters.

Many of those cost-savings have yet to be realized.

President Obama explained in response to a question from ABC News at a Sept. 2010 press conference that "bending the cost curve on health care is hard to do."

"I said at the time, it wasn't going to happen tomorrow, it wasn't going to happen next year. It took us decades to get into a position where our health care costs were going up 6, 7, 10 percent a year. And so our goal is to slowly bring down those costs," he said.

Obama added, "as a consequence of us getting 30 million additional people health care, at the margins, that's going to increase our costs, we knew that. We didn't think that we were going to cover 30 million people for free, but that the long-term trend in terms of how much the average family is going to be paying for health insurance is going to be improved as a consequence of health care."

Last year, I asked White House deputy chief of staff Nancy-Ann DeParle about that $2,500 in savings the president pledged. She insisted families will see that savings seven years after the president said it would be achieved - by 2019.

"Many of the changes in the Affordable Care Act are starting this year, and in succeeding years," DeParle told ABC News, "and by 2019 we estimate that the average family will save around $2,000."

DeParle said that the "big increases that occurred (in 2010) were probably driven by insurance plans overestimating what the impact would be and maybe trying to take some profits upfront before some of the changes in the Affordable Care Act occur."

Administration officials say they never claimed health care costs would go down, only that they would grow at a slower rate than they would have without the Affordable Care Act.

"It is clear that in recent years health care cost growth has slowed," Health and Human Services department spokeswoman Erin Shields Britt told ABC News. "In the ten years before the health care law was passed, health care costs increased at 8.7 percent per year on average, according to the independent Kaiser Family Foundation. This study showed a rate of increase of 3.8 percent in 2010 and 4.6 percent in 2011, while in a separate study, the Kaiser Family Foundation recently estimated a rate of increase of 4.0 percent in 2012, illustrating that overall health-care cost growth has significantly slowed in recent years."

Families are already enjoying savings through curtailed insurance premium growth under the law - compared to what premiums would be without it - officials say. They also point to tax credits and other provisions that are estimated to save middle-income families who purchase private insurance through new exchanges as much as $2,300 per year in 2014.

-Jake Tapper and Devin Dwyer

This post has been updated.

UPDATE: An earlier version of this story failed to credit John Merline of Investor's Business Daily with the computations cited above. ABC News apologizes and regrets the error.