If you're wondering how often you bust first, and then the dealer goes on to break afterwards, page 50 of Don Schlesinger's Blackjack Attack III has data from which you can derive the answer.
It shows that the dealer's bust percentage is 28.2%, while a basic strategy player's bust percentage is 15.9%. In general, I believe you can just multiply the two (.159 x .282) to find that you will have busted your own hand and lost first, only to find that the dealer broke anyway on 4.5% of all your hands.
This of course, is across the entire spectrum of player totals and dealer up-cards. You could get more accurate info on specific cases by multiplying your own chance to bust your total x the dealer's chance to bust her up-card. When you have 16 against a 7 for example, if you hit, you'll bust 61 times out of 100. Of those 61 busts, the dealer will go on to break 26%, or 16 of those times. So with that hand, there's a 16% chance that you'll both end up breaking.