No, I don't think so. You might apply to two or so, but no more than that--especially if they have application fees. I know the kids in my Interlochen classes definitely didn't apply to that many--in fact, few applied more than one place.
If worst comes to worst and you don't get accepted anywhere, you can always spend the summer taking college courses in creative writing and/or submitting your work to contests and publications.
EDIT: Ah, I see where HeavenlyJane is coming from. It looks like her daughter had a hard time with writing camp applications last summer. The thing about all the writing programs out there is that very few publish those lovely acceptance rates that everybody is dying to know. My advice would be to put together your portfolio using pieces that you love and that are unique. Don't write what you think the people reading your work will want to hear (i.e., very tragic stories to show your "serious side" or those poems with a bunch of random words to show your "deep side"...it's all happened

). At the same time, all artistic stuff is going to be subjective. After a certain point, you've just got to hold your thumbs.
If you do decide to apply to Interlochen and want advice on the portfolio, PM me or glassesarechic. We would be happy to help.