I've had 3 different careers and worked for 6 companies in the 19 years I graduated with my BA. Every job I've ever got I got through a connection and very proactive pursuit of that connection. And I have never worked in a job that had anything to with my major. Ding ding ding #2 is about ongoing education. I've always taken classes, kept up and added skill sets and done self study.
For those of you waxing poetic about graphic design, in 1999 I decided to change careers to graphic design and learned to code. I rapidly flew through a career that went from jack if all trades web designer to highly specialized information architect and user researcher. I am now a director of user experience at a Fortune 50 and I can tell you that we can't hire talent or retain talent fast enough -- people with GOOD skill in visual design for web, mobile or the tv screen/interaction design/information architecture/prototyping and psychology, cog sci or human factors skill sets. If you have a pulse and can document flows, wireframes and design or skin sites, apps and/or 10 foot experiences and you move to Silicon Valley with 2 years experience, you'll have your pick of employers large and small offering you 100K to start.
As for degrees, I used skills learned from my first two careers in marketing and sales to establish and sell in the specialized arm of the UX practice I now run - grew it from a team of one (myself) to a team of 8 in less than 5 years. My rock star team works at the intersection of technology, psychology, cognitive science and design. We all do the type of same work yet have degrees that range wildly: physics, advertising, communications, anthropology, sociology, information systems and graphic design. What we all share is a passion for detail and logic, insatiable curiosity, fascination with technology, need to constantly be challenged, ability to work and think independently to solve problems, and high work ethic. And what everyone who works for me has in common is that they came to me through a personal recommendation and proactively pursued the position with me.
And my personal learning never ends. While I run a UX team, I'm essentially an evangelist, coach, leader and manager and I actively seek out any and all opportunities to become a better communicator, presenter, mentor and motivator. I'm a long ways away from my original Physics BA or any graphic design class ... or even my graduate classes in human factors (a degree I didn't even bother to finish because job opportunities in UX are so great.)
Get a degree in whatever interests you, but more importantly, learn to network, get good at making small talk, practice being proactive, learn to negotiate and always follow through. These are the soft skills that will open the doors for a job and help you land one.