Bradford Pear trees are one of the worst. They're a small ornamental w/a genetic flaw that causes them to easily split down the middle. Most trees will break or fall if the conditions are right. Too much rain causing soggy ground; evergreens covered w/ snow; high winds. Or, if they're unhealthy & rotting.
There are plants that break their branches when they become too heavy. An example of such plants is the desert willow where too much water causes disproportionate growth especially after heavy rainfall. The branches of the plants therefore grow quickly and become heavy for the trunk to support, and therefore they break from the trunk.