WASHINGTON — Annual premiums for job-based family health insurance went up just 4 percent this year, but that’s no comfort with the price tag approaching $16,000 and rising more than twice as fast as wages.

The annual survey released Tuesday by two major research groups served as a glaring reminder that the nation’s problem of unaffordable medical care is anything but solved.

Premiums for a family plan are averaging $15,745, with employees paying more than $4,300 of that. And lower-wage workers are paying more for skimpier coverage than their counterparts at upscale firms.

Overall, “it’s historically a very moderate increase in premiums,” said Drew Altman, president of the Kaiser Family Foundation, which conducted the survey with the Health Research & Educational Trust.

He quickly added: “But even a moderate increase feels really big to workers when their wages are flat or falling.” General inflation rose only 2.3 percent, by comparison.

After a 9 percent hike in premiums last year, the 2012 increase quickly became fodder for the political debate. Republicans said President Obama’s promises to control health care costs ring hollow in light of the findings.

But the most significant cost-control measures in Obama’s law have yet to take effect, along with the president’s big push to cover the uninsured. The cost controls include a new tax on the most expensive insurance plans and a powerful board to keep Medicare spending manageable.

Trying to head off critics, the administration issued a report estimating that consumers have saved $2 billion as a result of the health care law. That’s due to a combination of insurance rebates for employers and individual policy holders, as well as closer state oversight of proposed rate increases, facilitated by Obama’s law.