I tent to agree with everything
steveash posted above. Determination is a good thing. However, if you are only an average designer after you graduate or do not have great communication skills and a 'big brain', you will probably struggle and determination may not be enough. I freely admit to being pretty average at writing and this has probably held me back.
Your success will also depend on the discipline you choose to work within. Digital design has obviously grown exponentially in the last decade and will continue to do so. So, do you choose to join a growing area? But also join thousands of other wanabee's? Or do you try and stand out from the crowd, should you be able to.
Experience is key and there will always be the 'chicken and the egg' scenario: without experience, companies won't want to take a risk. Unless you get a break by being determined, different, and of course highly skilled, it could be very hard to make a career in graphic design.
For a start it's not very well paid. I've 22 years experience and been freelance for 4 years on and off, in London (which as I'm sure Steve will attest to) is a global centre for design, even then work can still be hard to come by. (Having Top 10 "named agencies" on your CV sometimes helps).
I've been lucky enough to work as Senior Designer for Sony Design and Creative Head at the UK's No.1 design agency, Imagination. Not in digital design but in branding, experiential and packaging. Having that experience undoubtedly helped me and continues to do so.
When I graduated, I sent my CV and covering letter to every design agency in my locality and got maybe 3 interviews from 30 contacts. Remember, this was before email, so I couldn't even send a PDF of my portfolio. But I had something that was
needed at the time. Apple Mac experience. (this was 1990).
Like Steve, I started in a small presentation and design studio, on minimum wage. It may have been in a medium sized town, but it was the break I needed, with real work and real life experiences.
I worked my ass off every day, learned as much as I could on the job and after 1 year was head-hunted and moved to a larger ad agency in London. After that, my career grew and grew.
Any opportunity to show people what you can do is vital. Even if it's for 1 or 2 weeks and for no money. Some interns might get paid to cover their expenses, others will receive nothing. Do not be put off.
Suffice to say, a career in design is possible and enjoyable! But, if you are seriously concerned about money, especially after accrueing debts studying, you might want to think twice.
How about law?