I have no source to confirm this, but this is probably right.
Apparently, when you engage in a battle with Xerneas/Yveltal, regardless of whether you defeat it or catch it or whatever, a new Box 31 will be created in your PC.
So, really, there is no way your entire PC can be full of Pokemon, the captured Xerneas/Yveltal will just go to the newly created Box 31.
Hope I helped. :)