It's absolutely possible. I have an art degree and what I have found to be important is to have some impressive personal projects to show employers. Maybe not all of them will look at your portfolio but enough do that you should be able to get a job if it's good.

Falling in love with programming has been very fortunate for me because it is one of the few professions where you can show your skills with a portfolio instead of a piece of paper, and it pays pretty well