When I say full box, I mean all 14 boxes are full.
It doesn't even let you throw the ball. "The BOX is full."
That wasn't exciting.
For those wondering how I filled the PC but are too lazy to read the rest of this topic...
I have one of each of the 386 (except Lugia and Deoxys).
That leaves me 36 spaces. About 15 of those are filled with double rares. Another Suicune from Colosseum. Another Articuno from my brother. Stuff like that.
The last 21 spaces are normally empty, but I've been trying to get a Shiny Gardevoir. I just put all the Ralts that hatched from eggs in the spots to see what would happen.