I wonder if they meant that you shouldn't pay providers until things are sorted out. I agree that you need to pay your premiums, but doctors and hospitals will just send you to collections. Still not a great option, but it does keep the money in your pocket until things are sorted out. (Of course, I wouldn't do this unless I was very sure I was right, also, so that any collections issues would be an insurer/provider error.)
It is (also) their problem. If they're in the network, they've agreed to certain terms as well.
I'm not saying skip out on your bills. I'm saying that the provider and insurance need to figure the bill out. I'm not floating a provider in my network so that I can fill out a ton more paperwork to get reimbursed eventually.
And yes, they do make errors in major ways. Just two years ago, I got as though I hadn't made my deductible. It was a $2500 difference. It's much easier to have a provider refile to get their money than for me to pay them and then work through the insurer's system to get reimbursement.
As far as your provider is concerned, whether your insurance should pay or not isn't really their problem. I have never been somewhere that did not make me sign a form saying that I agreed to be responsible for whatever my insurance did not pay.