Atleast if it is a 'parallel' system which I seem to read here.. (sandwich) and the way you tell it..
There really is no way to mitigate this assuming all the firewalls are protecting the same systems. (without a really messed up architecture anyhow) Since each firewall would process all/nearly all of the traffic.

So catch, assuming that the school's firewall that I'm behind sucks (which it does), then are you saying that could make my computer more vulnerable than if I were only behind one good firewall?
You want to have the exact minimum controls to meet your needs and nothing more.

cheers,

catch