I just saw a clip of HHS Sylvia Mathews Burwell's claiming people support Obamacare because because, when asked, they claim they support many of the individual parts of the law. (I paraphrased her)
In order to understand this misconstruction, let's apply it to a different scenario. A family of three are asked if they would like to have a new vehicle. Mon says yes, she really needs a SUV for travel in snowy weather. Dad says yes because his truck is old and he is afraid his repair bills will start piling up, the son says a new motorcycle would be really cool and he could drive it to college without buying much gas. Democrats would say the family supports buying new vehicles when you ask them about it. Along comes a Republican who points out, mom will have to take a part-time job to help make the payments, dad will have to sell his bass boat because he can't afford three new vehicle payments and a boat payment and the son is told he will have to delay college by one year because mom and dad are tapped out on car and motorcycle payments. The Republican asks the family if they support the whole deal; including the consequences. Mom and dad immediately come to their senses, but the son is conflicted. He really wants a new motorcycle that someone else is going to pay for, but he understands the hardship this will cause his parents.
Now, let's move to Obamacare. Mom loves the free mammograms, birth control and subsidized coverage for her uninsured low income friend, but hates the fact she is losing her trusted normal primary care physician. Dad likes the removal of the lifetime limits on costs, but hates the higher deductibles and premiums. The son really likes being able to stay on his parents plan until he is 26, but hates the idea that after he reaches that age, he will be forced to buy overpriced health insurance coverage the rest of his life, even if he doesn't want it.
As you can see, cherry-picking the parts of any deal is a poor way to judge if people really support the deal.This is why Obamacare is so unpopular even if many parts of the law are liked by those who would benefit most.