Billing definitely happens. And one procedure can trigger billing from multiple different providers. I had my appendix out a few years ago and received separate bills from the hospital, the surgeon, the anesthesiologist, and possibly something else (nursing?). I was able to get the hospitals portion written off ($~20,000 and I was unemployed and uninsured at the time) but no such luck with the other providers, and those bills and the collections agencies followed me for years until I could get them paid off.