I've not worked in the games industry before, but looking at it from the general software development industry, I'd say that if you're relying mostly on your degree to land your first job, it can indeed be tough. You'll probably have to settle for a job that is a substantially different from your ideal job, and then try to work you way up/over.

The further into your career you get, however, the less and less relevant your degree becomes. Your past experiences, your portfolio, your professional network, and the ability to let all the knowledge and understanding that you have hopefully acquired naturally shine through in your interactions with potential employers become way more important. And you can get a jump start on a lot of that by making games. A lot of them. By the time you get your degree, it might already just be a supplemental item on top of your portfolio and acquired knowledge. Getting involved in game jams, conferences, and competitions can give you a jump start on networking also.

Given that your objective is game design, I'd strongly recommend learning only enough technology to enable to you make as many games as possible. Knowing technology is good for a designer, so don't ignore it altogether. But working through numerous failed designs and the occasional successful or promising idea along the way will be far more critical to developing your designs kills, and to prove to others that you're a capable designer. Heck, spend some of your time simply paper prototyping, or designing board, card, and other non-digital games. Maybe design a new sport. A lot of the knowledge and insights gained will carry over to digital games, but the pace of development can often be so much faster.

I just posted this blog post for someone else earlier today, and it might be something for you to consider also: "Game a Week: Getting Experienced at Failure".

Obligatory link to Tom Sloper's FAQ about getting into game design:

http://sloperama.com/advice.html

Game design and game programming are different things. Since you mention some C++ and a CS degree, I'm going to interpret that as a game design as the goal, but game programmer as the method of breaking in. Programming is a fairly reliable way to break in, as are the art disciplines.

It's not uncommon for game design degrees to include some programming work as well, e.g. DigiPen's BSGD program. This is relevant to actual jobs. Many shops greatly prefer designers who can at least write basic Lua/Python/C#/JavaScript/ActionScript beyond just "can tell the difference between a loop and a variable."

Designers with only a little script experience lead some shops to draw hard lines and ban designer scripts (because the scripts are so bad they do little but waste an engineer's time fixing them vs just having written them correctly in the first place) though others have started to explicitly want more technically-capable designers so they can iterate on crazy ideas and small gameplay features on their own (while engineers spend their time on tools, architectural/support code, performance, and major/complex gameplay systems).

People tend to forget that not every game designer is the guy who comes up with the main game theme or primary gameplay schtick; those come about via collaborative processes and not some rockstar designer's grand vision. Most game designers work on specific piece of gameplay, enemy design, level design, bits of story, etc. and are just one small cog in the machine (the same as engineers, artists, producers, etc). If you are a designer with a CS background, you have significantly more value than just being someone with ideas (if that's all it takes, everyone is a game designer).

Short version: designers can benefit from CS degrees for good reasons besides just getting a foot in the door.