Getting a Bsc(Honors) in Computer Games Design isn't a bad idea, but it's obviously no guarantee for success in the industry at all. In fact, mostly it doesn't work like that at all. There have been many complaints from the industry that many of these courses don't teach the necessary skills needed at all.

Personally, I think specializing is actually the best thing to do, as that's what's needed more and more in the industry. Games are becoming so complex that you need to be able to work in teams and do very specific (programming) jobs. Frank is right though, getting into the designing part of the industry is probably even more difficult,

Cheers