I bought a no-name-but-not-bad wrench set when I was a teenager (this was ummmm 30 years ago), and still have it, and I've never broken a wrench. You're not likely to, working on bikes. Maybe if you're working in a bike shop... though these days, you'd mostly be stripping allen keys, I guess.

If you also work on "heavier" stuff like cars, then yes, maybe you want to worry about what to do when you break wrenches. But I've never broken a wrench working on my car, either, though I know it's not unusual.