There is indeed enough space; you even have a couple of boxes left, IIRC. Definitely not enough room for shinies...if you wanted to do that, you'd need Pokemon Bank, which has an insane amount of storage space. But never having to go after the more annoying Pokemon again is a definite plus...the only real downside is that unless you move everything over, you'll never want to play the game again. xD
Yeah, and they're sending me the broken copy back, as well, after keeping it for about two months. Nintendo Support aren't the most brilliant of services; they could have at least offered me a discount or something. It'll be interesting to see if they respond to my e-mail or not. xD