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Workers saw a modest rise in the average cost of employer-sponsored health insurance this year, but they’re probably not overwhelmed with relief.

Coverage costs still are climbing faster than wages. That means, in many cases, a bigger portion of the average paycheck is sliced off for insurance instead of being deposited into bank accounts.

Annual premiums for employer-sponsored family coverage climbed nearly 4 percent this year to top $16,000 for the first time, according to a survey the Kaiser Family Foundation released Tuesday.

The cost of single coverage rose almost 5 percent compared with 2012. Those are smaller increases than the spikes of 9 percent for family coverage and 8 percent for single coverage recorded in 2011. But this year’s increases lap a 1.8 percent rise in worker wages over the same period.

Plus, more companies are giving their employees coverage with a higher deductible, which requires a patient to pay more out of pocket for things like blood tests or MRIs before coverage starts. Coupled with the growing cost of coverage, that means some employees may be paying more for insurance that covers less.

While health care costs have generally grown more moderately since the Great Recession eased, the average worker still feels the pain of paying more, said Drew Altman, CEO of the nonprofit Kaiser Family Foundation, which conducts the survey on coverage costs with the Health Research and Educational Trust. “Their costs are going up, their cost-sharing is going up, wages are flat, and inflation is much lower,” Altman said.

Employer-sponsored health insurance is the most common form of coverage in the United States. Employers typically cover most of the health insurance bill for their workers, and the actual change a worker sees in health insurance costs can vary greatly.

More than half of companies with fewer than 200 employees offered insurance with an annual deductible of $1,000 or greater this year for single coverage, according to the Kaiser survey. That’s up from 16 percent in 2006.

Expect high-deductible plans to grow. A total of 38 percent of all workers with single coverage have a deductible of at least $1,000 this year. That compares with 10 percent in 2006.

These deductibles can help employers avoid an overhaul-mandated tax on expensive plans that starts in 2018, said Paul Fronstin, an economist with the nonprofit Employee Benefit Research Institute who wasn’t involved in the Kaiser study.