I graduated from college in 1999 with a BA in Communication but I had been making web pages with static HTML for four years and had done some small freelance sites for companies in my home town during my summer breaks. I found my first few jobs very quickly, trained myself in some new skills, and was offered a promotion into the "engineering" (aka "real programming") department very early on. It is definitely possible!

At the time, I made a "portfolio" with color printouts of the web sites I had designed and built, and attached it to my resume. I believe that was the strongest element in getting hired: I had already shown them that I could do the job. Maybe your github account is the 2014 equivalent of my portfolio. It depends what you have on there, and who's reading it. I would also consider providing them with compiled, working applications, because the gatekeeper might be an HR person who doesn't know how to evaluate source code.