I work for a huge insurance company so here's the 411. Your insurance company will not notify you when a doctor drops out of their network. Most insurance companies keep a database on their website with their providers listed. Doctors and join and drop out of networks at any time so you really do have to keep on top of things.
I'd say that you should check with the insurance company to make sure the bill was submitted correctly with the proper dr's name listed. Doctor's use billing services to submit their paperwork and the billing services can make mistakes. If the paperwork was submitted correctly then ask the insurance rep to check to make sure that the provider really is no longer in the network. Ask them what date the dr dropped from the network. If it was close to your appointment then possibly the office staff hadn't been informed yet.
Once you have this information contact the doctor's office. If they submitted something incorrectly ask them to submit a corrected billing. If they are truly out of the network ask why you were not informed of thi when you made the appointments since they verified your insurance info. You can certainly ask for a discount on services due to the lack of info which they may or may not give.
The part you may not want to hear is that you do have to pay for the billing. You signed paperwork at the dr's office saying that you are responsible for payment for services rendered. If you choose not to pay they can turn the billing over to a collection service and ding your credit report for this. If you can not afford to pay the billing in full set up a payment plan with the dr's office of say $5-$10 per month to pay it off. As long as you are making payments they can not send you to collections.