If I have a pick with Fortune II on it, could I later apply a book with Fortune III to it and have the Fortune level increase to III? Same with other enchantments; can I “upgrade” items’ enchantments’ levels with enchanted books?
Yes, in fact you'd only need a Fortune II book (or another pick), not III. If you combine two items with different levels of the same enchantment, the new item will have the higher level of the two. If both have the same level, the new item will be one higher (unless they were maxed out already.)
For more info, see the wiki page on Anvil Mechanics.
Village Mechanics: A not-so-brief guide - Update 2017! Now with 1.8 breeding mechanics! Long-overdue trading info, coming soon!
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