We should all be familiar with our own insurance policies and in particular how pregnancy and delivery is handled. I have found that the hospital's finance office is usually very helpful in estimating costs based on my insurance, regardless of the type of surgery or procedure. They have elaborate lists detailing typical costs.
With my last pregnancy I was able to figure out in advance how much my c-section co-insurance would be, as well as the typical costs for my newborn (don't forget about those -- since he/she is a different patient at that point, the baby gets his/her own bill). This was I was able to guesstimate how much to add to our flexible spending account to cover these costs, especially since she was born in November and we had a "family status change" and could legitimately change our FSA numbers.
My previous doctor, as well as my current one, charge a "global maternity fee" that covers a set amount of u/s, blood work, and visits. They submit the charge to the insurance company around the 7th or 8th month of pregnancy and then set me up to make payments on it starting then, too. This time around, I'm due in Feb. so I had asked early on when I would owe money. But the clerk told me it was only after the baby was born. Well, at my last vist (20 weeks) I was asked to start paying! I told her I'd be happy to pay in January, but since we'd already spent our FSA for this year, I'd really rather wait. Plus, I'm due at the end of Feb, so that's a good two months of time to see my smiling face in their office to hound me for money.
And remember, ladies, it's always more expensive if you have a c-section... so ask about those prices, too. Oh, and since you're probably paying a percentage of the total price that is then knocked down by some other previously-agreed-upon percentage, I always ask for a detailed bill to ensure that it's accurate. My daughter's latest eye surgery had some funny stuff on it, so I asked for it to be audited, and it dropped by $2000!
There is an insurance board here on Healthboards.com, so that's probably a better place to have this conversation, since this isn't technically medical related.