This definitely seems weird. I go to Progressive Care for Women, affiliated with Northwestern, and had both my babies at Prentice (ages 4 1/2 and 16 months). I was only required to pay a co-pay on my first pre-natal visit for each pregnancy, then paid nothing after that until it went through my insurance. After the invoices were processed by the insurance company and my OB was paid by them, any remaining unpaid balance was billed to me by my OBs office. I did not have to pay for their actual births until probably 6 weeks after they were each born, because that's about as long as it took for the insurance company to process.
I should note that I had Aetna PPO with each child, but I don't think that should really matter. How can you be billed for a service you did not incur yet??? And who knows what might happen in the delivery room - what if you need a re-dose of your epidural? What if you need an emergency c-section (hopefully not!)? How could you possibly predict those costs at this point? I don't know...it all seems strange to me.