I've heard some negative things about game design degrees. Some of them are too new, some of them don't adequately prepare you, and some of them are outright scams taking advantage of the enthusiasm of their students and leaving them with a completely worthless degree and no skills to back it up. Perhaps they've gotten better recently, but because of what I've heard, I wouldn't recommend them.
That's all hearsay though. Perhaps there are some that are good, but it's very hard to tell until you've actually gone through the program and thrown a lot of money at it. Doing a quick google search finds people complaining about ITT Tech, but it also finds people defending it. Were the people who were complaining simply not putting in the effort? Did they get a bad teacher? Or is the quality of ITT Tech's education subpar? I can't tell you.
Most professionals I've heard talk about them recommend avoiding them, and recommend getting a traditional programming degree instead, as that can give you skills that are valuable outside the game industry as well, which can be helpful and give you more options for work as programmers are needed everywhere.
If you're doing independent development though, there's also the question of why you would want one at all. If you're planning on making your own company, then why would you even need a game design degree to do that? You can learn how to make games on your own and save yourself the money. In my opinion, the best way to learn to make games is to just get started trying to make games. That and studying the masters. There's also a huge amount of information out there on the web. Gamasutra.com is a great place to start.
I'm not saying you shouldn't go to college - rather, I don't think you should go to college specifically to learn how to make games. Go to college to learn how to program or make art or some such. It'll give you much more options in the long run, and you can probably do a better job teaching yourself how to make games anyway.
The best advice I can give you though is to have a day job while you work at the game you want to make. Don't put all your eggs in one basket. That way if you fail, it won't hurt you financially, and you can try again. Ask around in lots of different places where industry professionals reside about these degrees and what the best path is to take, like maybe forums.indiegamer.com or some others. Get lots of opinions from people who would know. And don't cover your ears if someone is critical and tells you your work could be improved somehow, because even if they're rude, and while some of those people should not be listened to, if your work really could use improvement, it's an opportunity to improve, which is something you should never stop trying to do.
Edit: Oh, and for crying out loud, start small, and unless your dream game is small, don't make your dream game first! You dream game will be much, much better if you wait until you know what you're doing a bit better when you make it!