The number one rule is don't pay anything until it is resolved. The second you've made payment, nobody else cares to make any effort.
Pay on time, keep documentation you paid on time. Hell, turn on autopay and keep documentation on -that-.
Don't ever think you can hold your premiums hostage. You'll only be screwing yourself.
Edit: if you're referring to clinic bills, be sure you're in active communication with the practice manager so you don't get sent to collections. If it's a hospital, you'll be sent to collections anyway - they automate that process, once you're sufficiently late.
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.