I wish I had grabbed a screenshot, but basically what happens is this:

If the inventory is full (say 230/230), the game will not allow you to add items from at least hoards, possibly other areas even if these items stack (oatmeal, for example).

If the total number of "stacks" exceeds your inventory max, the same will happen. For example, if I have 227/230 slots filled and 4 items in a hoard, the game rejects putting these items into inventory with the "add all" button and states the inventory is "full", even if say 3 of the items are stackable. Added one at a time, it works, and the player will end up with 228 slots filled.

This is assuming the stacks are not 99x items, which was true when I encountered it.

As GreySkyMorning said pixpet.net/forum.php?section=posts&t=1285&title=Forum---Site-Bugs---Unable-to-add-same-type-of-item-to-full-inventory

But...I can see this being brought up again by players in the future who don't know that it's intended, because it looks like something that's broken. In most games with inventory management, you can add to a stack that isn't full, regardless of the condition of the rest of your inventory. It's just logical. So I'm wondering if there's a remedy for this that won't affect the server. And if not, maybe there could be an error message that explains that it's intended and why. Otherwise people may continue asking about it.