Employees have always bellyached about the quality of their health benefits — it's practically a rite of passage in most mid-to-large-size American companies. Now, a steadily increasing number of workers — 20 percent, compared with just 10 percent in 2012 — say they would rather make more money and have fewer health benefits, according to a study in the March issue of the Employee Benefit Research Institute's monthly newsletter.

Yet that doesn't mean they're totally unsatisfied. When push comes to shove, the EBRI study indicates, most workers are at least somewhat confident their employers have chosen the best option for their coverage.

With higher pay but fewer benefits, these employees could, theoretically, take that extra money and use it to purchase those extra benefits on their own. But in the same study, a greater percentage of those surveyed in 2015 (50 percent) than in 2014 (47 percent) indicated that they'd stick with the coverage currently provided to them.

In fact, the number of people who would prefer to just shop for their own insurance has steadily decreased — by 10 percent — from 2011, a year after the signing of the Affordable Care Act, through 2015.

The same goes for the number of people who would like their employer to hand them the money earmarked for their health insurance so they can make their own decision in the free market; that's down 7 percent in the same time period.

The EBRI attributes this decrease to basic trust in the people responsible for choosing employees' health plans — or, perhaps more accurately, their lack of trust in their abilities to do it themselves. And despite a slight dip in the percentage of people who consider an employer offering health benefits as either "extremely important" or "very important" over the last year (80 percent in 2015 versus 82 percent in 2014), for the vast majority of workers, employer-backed health insurance is still a must-have.