So why do wood fence posts rot in this way?
When it rains, the ground becomes saturated. The concrete base, the fence post has been set in, will protect the below-ground wood of the post, but the wood at ground level just above the concrete is in constant contact with the soil and is therefore unprotected from the elements. Over time wet soil will cause this part of the wood to rot. As the rot worsens, the fencepost becomes weaker and eventually a strong gust of wind will cause it to snap.
Better to repair the broken post soon. Before the fence nears total collapse!