Yes, it is a problem - an 8Mb service is insufficient to support a whole class making extensive use of the Internet, and all connectivity dropping out for 10-60 seconds is unacceptable.
I haven't yet tried isolating everything in the way you suggest, as I can't take really do that until the Easter holidays, so I was hoping to find something sooner than that.
Have you had a chat with management about it? You may find them supportive of a little downtime to resolve/isolate the long term issue that is impacting on T+L. If there are two of you with radios/phones it shouldn't take more than an hour to isolate the device(s) causing the issue.
The trouble will be if the problem is intermittant - in which case you might not be able to tell whether you've isolated it or it just has gone away on its own, which would be a waste of everybody's time, something you are obviously keen to avoid.
The next thing i would do is to look for packetloss on your internet pipe. You ISP can give you this information or if you have a decent router then you can check that info on the interface that connects to your ISP.
Build yourself a dual homed firewall that will log all traffic.
You can use this to segment areas of your network.