When weapons and/or armor run out of durability and enter the "Broken" state, repairing it in an anvil doesn't take it out of its broken state, but repairs durability.
You can see in the screenshot below that the Leather Cap is fully repaired, but stuck in the "Broken" state and is unusable.

The only way to fix it is by using NBTExplorer and editing the item in the player's inventory. Which gets incredibly tedious when you have to do it every time your armor breaks.