My wife of eight years gave birth to our first child on June 15th. I had saved $2,000 in different banking accounts for "emergencies." Our company health insurance is a high-deductible plan. Meaning, we pay out of pocket up to $4,000 before we ever get anything, then it's another $2,000 out of pocket (at an 80% discount) before we hit a 0-out of pocket limit. We still haven't hit that this year (or ever, actually) but we have come very close with this latest experience.
The hospital bill for my wife and child (after insurance) was over $2,300. I cleared out both accounts and spread the money over the hospital, clinic and anesthesia group that was for some reason, a separate entity from the hospital. All totalled, our "final" bill for just the birthing event was somewhere in the $5,000 range. But, paying $2,000 split across four bills (they wouldn't combine my wife's hospital bills with our son's bills) and setting up payment plans for everything else has helped keep things in-check.
One of the first things we did was to adjust our tax-withholdings from single employed one dependant to two dependants. This increased what I brought home every pay check. We then took a hard look at our monthly bills and reduced our spending on eating out and even cut out our cable subscription. I'm doing better with unplugging unnecessary electronics, leaving windows open on nice days and the AC/heater off when we can. We actually have used less energy and lowered our bills substantially.
While we're not totally free of the medical bills, we're on track to be totally free (without charging everything to a credit card) before the end of the year. We will continue our low-cost living expenses and start putting the money we save back into the empty accounts to rebuild for the next "emergency" whenever that may come. Once those are replenished, it's on to saving for college and paying off the house early!