Cambridge maths, good grades, bad job prospects

Hey guys, first post here, but I'm a long time reader.
I am in my final year (but will do a 4th year if I land an internship offer) in the University of Cambridge, doing mathematics and although I've had only one rejection yet, I am fairly sure I am not getting past the CV screening for any of the summer internships I am applying to (I've applied to most of the BBs and 5-6 other banks). I've been to careers advisers in uni (they say my CV's fine), I am part of my college's investment fund and have been pitching stocks, I have good grades at exams, I have been going to careers events since my first year, I was in the committee of a 100+ member boat club and still - nothing.
What can I possibly be missing? I am not even that picky when it comes to choosing a division - I don't want to do anything boring, but still I'm not only going for the most competitive divisions either.
The only companies which have interviewed me are software companies and although I'd love to be a programmer for some time, I am pretty sure I'd get bored with it within 5-6 years tops and I have been dreaming about a finance career since high school.
So can you just give me any sort of general advice on what can I be doing wrong and how to fix it? Also is it possible to work in an unrelated industry and potentially land a job in banking later on?