tl;dr - Stick at it
Normally a studio won't look at you without 2/3 years experience, the sheer speed that work comes in and flies back out again is enough to overwhelm anybody and small mistakes can be costly. Get your portfolio together, start working for friends, work for free if you have to.
I've been questioning the higher educational system for a while now, especially when it comes to New Media. I spent 6 years studying Film and Television at College/Uni. Pretty hardcore production stuff; left with a great portfolio, I'm actually a pretty good camera op (which is what I wanted to do). Our lecturers from day one made us all think we'd float out of uni, straight into camera assistant jobs, assistant editors etc, just to fill course places. Probably a handful of my friends actually managed to get jobs related to their study. After 4 years, my housemate is still doing the crappy freelance grip/assistant to the camera team jobs that aren't regular. He found out after leaving Uni that he was going nowhere without at least 8-9 years experience. In short, he could have skipped Uni entirely, he'd still have been starting at the bottom. It's all about being in the right place at the right time, getting a job at the bottom (in house is where I started - eww) and moving up.
In short, don't give up, get a job that supports you and spend all your spare time sharpening your skills, learning new things and getting some experience. You'll be taking your name with you from job to job, employers know that their business is only as good as their designers. There are loads of people coming out of Uni with degrees now, it's no longer a way to stand out.
If you're serious about getting a job as a Designer, it's a complete lifestyle change. I work at a studio, then when I get home I work on my own clients, in my spare time I'm out trying to sharpen my photography skills or add another skillset to my portfolio. It's a constant, life consuming task trying to stay on par with the thousands of talented designers out there.
Don't give up though. And get a portfolio online or you're going nowhere!