Quote by ballzac
Firstly, wow! You really went above and beyond give me a thoughtful and comprehensive response, so thank you very much.

No problem.
Secondly, I meant to post this in the quantum forum, as that is what I am studying at the moment. Of course having understood the majority of your post, I also see that this will hold for the quantum case where it is continuity of probability current that is being being described by the equations.
So, if we were to make a measurement that violated the continuity equation, could we also say that gauss' divergence theorem has failed?

Absolutely not. Empiricism has no place in judging the validity of mathematical propositions!
Rather, it would testify either of two things:
a) The measurement was false
b) The measurement is correct, but the conditions required for the appliciability of Gauss' theorem are not present. (I.e, you have a case of misapplication)
Which of these would be the right one in your particular case, I don't know.