Doesn't the at-fault driver's insurance company have to pay my medical bills as they are incurred?

A: No.  This will come as a suprise to many people involved in a personal injury case, but the insurance company for the person that injured you will not pay your medical bills as they are incurred.  They will pay you one time at the end of your treatment.  This payment is designed to include ALL of your damages including medical bills, lost wages, and pain and suffering.

For this reason, unless you have medical payments coverage or good health insurance, it is frequently very difficult to get the medical treatment you need following an accident.