This is inevitably what happens when the people reporting the issues get to decide their priority (I've seen this even in a two-person dynamic). You have to have discipline, and you usually have to have an impartial outsider (to the issue at hand) who doesn't care about other people's panic to actually set priority levels.
I've seen systems that have two fields, e.g. priority and severity. Where severity is "how important the client thinks it is" and priority is "how important we think it is".
Our system has another one: Who it impacts. Does it impact a user, a department, or the entire organization? The difference between "my crap's broken" "our crap's broken" or "everybody's crap's broken".
