We are lucky to have good health insurance, and a simple ER visit to get stitches or an X-Ray will invoke 5 pieces of mail from 3 different entities. You have to watch out for student Doctors who will send you a bill for being in the room. It's too much to think about when you should be focused on taking care of your health issue.
This is one of my biggest pet peeves. The biggest bill I've ever had get processed wrong is the only one where lawyers ended up getting involved, and what the other side's lawyers sent was a form my wife signed while in labor (was at an 8 when we got there) while I was out of the room agreeing to pay the bill however the hospital decided to bill it. They said that excused their mistake, and now that it was too late to bill insurance (had been over a year) we were on the hook.
For example, an Anesthesia resident will perform all of the procedures and monitor the patient's vital signs and response to the general anesthetics or other drugs. The Attending physician only intermittently checks on the case if the resident needs help or the case becomes too complicated.
If the resident doesn't bill for services performed, such as central line, placement, peripheral line placement, intubation, arterial lines, etc. I'm not sure who would do the billing.