I work in healthcare IT, and eat lunch with some of the guys who maintain our billing systems. I don't understand the whole billing process (it seems like black magic), but I know a bit about how it works.
The insurance would have been billed for both services, and they will check that all the paperwork is in a row before sending payment. You generally will not be charged any of your portion until after the payment from the insurance company has cleared. Usually that's within 30 days of the service. If, however, the pre-authorization number wasn't entered right, or the insurer didn't think the provider picked a specific enough ICD-9 code (out of 13,000 codes) they'll deny the claim, and the bill has to be re-sent with updated documentation, etc. They may request an audit of the whole record. I've heard of those bouncing back and forth for up to 2 years. Then it finally gets through insurance (with lots of room for human error), and they come back and bill the patient. I can imagine a scenario where the insurance company thinks it's in the hospital's court (so they show nothing outstanding), and the hospital takes 2-3 weeks to gather additional documentation to justify their charges.
I would check to be sure which date of service (September or November) the bill is on, and double-check that you didn't accidentally get billed twice. Check with every party involved (anesthesiologist, surgeon, hospital, insurer, etc.). Be kind, but require full and detailed documentation of exactly what happened, where all charges came from, and what was paid. If something was hanging out in limbo for a full year they should be able to provide you with details as to how much and why.
Good luck!