<p>I once was asked to increase thread priority to fix a problem.  I refused, saying that changing it was dangerous and was not the root cause of the problem.</p>

<p>My question is, under what circumstannces <em>should</em> I conider changing priority of threads?</p>

Not many.  The only time I've ever had to change thread priorities in a
positive
direction was with a user interface thread.  UIs must be extremely snappy in order for the app to feel right, so a lot of times it is best to prioritize painting threads higher than others.  For example, the Swing Event Dispatch Thread runs at priority 6 by default (1 higher than the default).
I do push threads down in priority quite a bit.  Again, this is usually to keep the UI responsive while some long-running background process does its thing.  However, this also will sometimes apply to polling daemons and the like which I
know
that I don't want to be interfering with anything, regardless of how minimal the interference.