I've caught a lot of Pokemon I wish to keep, but I'm starting to become concerned I may run out of room in my PC. Is there a way to add more boxes, and how?
Like in Gen VI, place at least one Pokemon in every box you have and log out of the PC. Go back in and you will have an extra free box. Continue filling your PC Boxes and logging out and eventually you'll max out at 32 PC boxes.
Hope I helped. :)
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