If your fly rod doesn't get broken at least once, you aren't using it enough.
Fly fisherman can be hard on their fly rods. Most fly fishermen aren't super hard core. Weekend warriors who fish once or twice a month don't break many rods. However, you also have those really crazy fly guys that fish through any conditions. Well, those guys break gear. They hook into bigger fish than they should with a fly rod they know they shouldn't be using. However, that's part of the thrill of it all.
With that in mind, just how much stress can a good fly rod handle? You might be surprised.
One of my favorite carp rods is a 5-weight Sully from Walton Rods. I fished all summer with the prototype of the finished version, which is due to be released in January 2018. I also used that rod to handle king salmon and steelhead in Michigan and New York. Yes, a fly rod can handle a lot of abuse from big fish.
Often times, though, it's better for the fish to use gear more suitable for the fish you are going to be chasing. It reduces stress and prevents break offs. However, you can't stop some fish from hitting your flies and sometimes things just happen.
Sometimes you just need to have luck on your side.