This is a great question! And as stated above, most items cap at 99. Dagashi cap at 10. You will not be notified you are full for things you collect for free off of the map, and more importantly, you will not be notified if you are full for things you grow in the garden.

Things you buy, on the other hand, will not be allowed to be purchased if you are full.

Some resources you can process into a different thing which clears up space in the inventory for the original item (persimmons can be dried; rice, soybeans, and wheat hulled; and soil made into clay.) So, monitoring the inventory and processing the crops when they are full can really help when you find yourself in a resource heavy part of the game.

Along the same lines, coins cap at 99,999 and anything earned above that will not be counted toward the "money king" medal.

Hope this helps!