I recently got a $5K hiring bonus at work when a friend of mine joined the company. They double-paid me. I notified payroll, who acknowledged the problem and said that they'd take care of it. It's been over a month, though, and it hasn't been handled.
Just before posting that poll I remembered it. "Oh yeah, I should nag them about that." That sent me down the path of pondering such errors, and where the ethical boundaries were for letting it slide.
For me, the impact that the mistake will have on the erring party make a huge difference in how I handle it. If a large entity makes a small error in my favor, I'm unlikely to care much. Like most of the respondents, I'll try harder if it's a large error, a small entity, or someone that I regularly do business with. I might not try at all if my bank made a one dollar error in my favor. I'd definitely point it out if Chop Bar (the restaurant in my building) forgot to charge me for a cocktail and I noticed, but I might not do so for a glass of iced tea.
JP pointed out a case of a large business entity making large mistakes in his favor. I similarly will not correct them in that particular case.
If it's my mistake, the equation changes somewhat. I try harder for those.
The question I don't know the answer to... would there be a point at which I just give up trying to return $5K to my company? Obviously, one attempt was not enough for me. If I try three times and fail, have I done enough? I don't know.