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.