I think this is a place where character shines through. I might stay for a few hours and seriously think about the issue. I try various avenues of attacking the problem but never quite solving the entire puzzle. Sometimes after this tiring search i come up with nothing. And then, in a fit of exhaustion, i go to the bathroom.
Your brain knows that when you go to the bathroom you can (and should) relax because the purpose of the bathroom itself is to get you rid of all kinds of residue (which can't happen effectively without relaxation) - your brain knows this and slips into the feedback loop which allows your mind to relax as well. In the case i describe above, idea by idea (almost involuntarily) i go through whatever i tried and whatever distracted me - and the mind discards some of this population of thoughts - because that's what you do in the bathroom. You discard stuff. The mind, now relaxed, just like a muscle, can 'lift more weight' and can try to tackle the problem from angles which might not even have been visible before because the search itself was exhausting it.
So i'm not a doctor but.. when you feel like going to the bathroom or taking a break.. maybe it's not a bad idea to actually do it :)