I started learning web related stuff from home, as a hobby. So I never did go to a official course on it. In the UK you'd be hard pressed to find an entire college degree in just web development. I find that Computer Science courses abound, and they teach a general sylabus of programming - usually with Java etc.
With doing something like Computing, it's a wide introduction into many professional careers. I think you can only compare it to other professions, such as Law, Business, Medicine etc.. at this point it becomes clear that if you are good with IT then stick to it rather than go with Law for example. However if you chose business and did IT -specifically Web Development- on the side then the obvious benefits can been seen. You should choose what you need to know most and what you can probably pick up/learn on the fly that will best meet your needs as you envision them in the future.