Hoppers directed into chests - What happens when the chest is full?

one thing that you could do would be to place a hopper under the chest leading into another double chest, or you could have a overflow protection device that burns/destroys extra items, to have the chest system vertically you could have chests like this:

Here are two types of "overflow" systems, each holds the same number of items and has the same footprint. If an item is tossed in the top hopper, it will flow down and be available in the bottom chest. This is true until the bottom chest and its feeding hopper are full.

With the system on the right, when items are removed from the bottom, the inventory from the upper chests will flow down to fill gaps. By contrast, when the bottom tier of the left system is emptied, the upper tiers will not fill it (though overflow held in the hoppers will flow down). This is because there are no hoppers pulling from the chests.