If you were in a car accident, chances are that the day care would waive the fee. The fee doesn't exist to be needlessly cruel, it's there to encourage parents to pick up their children before closing.

As far as passing out in your home for a few hours. If that were really an issue for you, then potentially paying $600 might motivate you to actually go see a doctor. Otherwise there could come a day when you pass out and never wake up (and your children never get picked up by you or see you alive again)...