If you have two identical pickaxes (same tier, same enchantments, renamed same) and one breaks, under certain circumstances you can use the other one to glitch it back into existence.

What I expected to happen was...:
A tool that has broken is gone.

What actually happened was...:
I replaced the broken tool in my hotbar with a nearly-new, identical tool, and the one that had broken dropped in front of me when I closed my inventory. I was able to break about 20 more blocks before it broken again, and I recovered it again. It broke a third time and I wasn't able to recover it.

Steps to Reproduce:

(1) Go into Creative Mode.
(2) Enchant a golden pickaxe with Efficiency V, Unbreaking III and Fortune III in that order.
(3) Rename the pickaxe.
(4) Duplicate the tool by middle-clicking.
(Originally, I got two identically-enchanted iron pickaxes by chance in survival mode. I named them both "The Twins" as a joke. I believe the identical names and three identical enchantments are key to the glitch. Not sure whether the enchantments have to be in the same order. Repair cost doesn't need to be the same.)
(5) Go into Survival mode.
(6) Break a few blocks with one pickaxe to give it non-full durability. (Not certain whether this is necessary.)
(7) Break blocks with the other one until it breaks. When the pickaxe breaks:
– You must be looking into air blocks (not certain whether this is necessary), and
– No stray item on the ground may jump into the hotbar slot where the pickaxe was. It must remain empty.
(8) Do not move your mouse after the pickaxe breaks. Open your inventory, click on the remaining twin pickaxe and place it in your active hotbar slot.
The tool that just broke may (1) reappear in its hotbar slot when you pick up the twin, (2) reappear on your cursor when you place the twin in the hotbar, or (3) drop onto the ground in front of you when you close your inventory. The recovered pickaxe will be at near-zero durability and will retain its repair cost.

Sometimes, when the pickaxe breaks, the tool-breaking sound doesn't play. If this happens, the tool can't be recovered.