Similar here. Had a baby 9 months ago, the doctor took the insurance, but it turned out later the hospital didn't. Bill was over 100K and it wasn't covered at all.
I asked the hospital who the anesthesiologist would be and they said they had no way of knowing because you use whoever is on duty when you give birth.
So there was literally no way to figure out if all of the birth would be covered. And it turns out, the anesthesiologist that happened to be working that day was not accepted on our plan. They billed us for $4k (even though he botched the epidural, causing us to need an emergency c section.)
We had to decline to pay it a few times before they said they would work it out and accept the lower amount from our insurance.