How do I make room in my inventory?

So I found this rad new axe that I wanted to equip, so I dropped my old one and went to pick the new one up.
Unfortunately I just got an "Inventory Full" status.
So I dropped a few more items from my inventory... and then a few more, and then a few more.
But whenever I try to pick up the axe the game still refuses because my inventory is full.
How much do I need to drop before I can pick up a new weapon?
Edit: The plot thickens. I tried dropping and picking up my Steel Sword and old Axe, and was able to reequip both without issue. Is there something special about the Temerian Iron Axe I'm trying to pick up now?
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How can I increase my inventory slot?
How to increase inventory space - Breath of the WildWhat do you do when your inventory is full in Minecraft?
It is now possible to have extra inventory slots (as of 1.11)....Instead, there are other ways to carry more:Ni No Kuni: Cross Worlds - FREE Inventory Expansion! Get MORE Bag Space FREE!
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