Decayed Chests are broken chests that must be restored with the Amulet of Uroborus, which is not acquired until the very beginning of Chapter 13 - Passage to Delphi. You encounter some Decayed Chests earlier in the game, but you can't restore them until Chapter 13. All you have to do is go up to them, hold L2 and then press Square to restore the chests. There are only 7 in total, with the last one found in Chapter 19 - The Foot of Apollo. Getting all 7 in a single playthrough will net you the If It Ain't Broke... trophy.
#1) Passage to Delphi - After you heal the bridge at the beginning of the chapter and can go right to speak with the Oracle, go left instead and you'll find the Decayed Chest at the end of the path.
#2) The Cistern (2) - When you enter the Cistern again and drop to the floor with the giant statue, you'll fight some enemies (Feral Hounds and Satyr General). After killing them, go into the hallway that is shown and you'll find the Decayed Chest right there.
#3) The Cistern (2) - From the previous chest, find the little alcove with the mysterious painting (the one with all of the letters). The Decayed Chest is opposite of the painting.
#4) The Cistern (2) - When you get outside, you'll end up fighting a Cyclops and some bugs. After killing them, look to the right of the doorway you came out of to find the Decayed Chest.
#5) The Cistern (2) - From the previous chest, you need to go to the left and use the Amulet to repair the wooden bridge. Instead of doing that, drop down and go all the way to the right to find the Decayed Chest across the small bridge that leads to the roller puzzle room that you did the last time you were here.
#6) Prison of the Damned (2) - This one is located near the end of the chapter, after the series of green grapple points. You'll go down some stairs and through a gate, where you'll find the Decayed Chest to the right.
#7) The Foot of Apollo - This is at the beginning of the chapter. After pushing the giant rock out of the way, there's a spot where you need to do some climbing. Once you reach the first platform, go to the right to find the last Decayed Chest.
