At a place I used to work they typical response to any problem was to blame the hardware or the users for not using the system perfectly. I had adopted the philosophy that it's my fault until I can prove otherwise prior to that job (and so far, at least 99 times out of 100 it's correct).
One of the last "unsolvable" problems when I was there was an abundance of database timeouts. After months of research, I still only had theories but couldn't prove any of them. One of my developers adamantly suggested replacing the network (every router, switch, access point) but couldn't provide any evidence that the network was the cause; it was, however, "obviously the cause" according to my manager (no development/IT experience) so he took over the problem. One caveat and Fog Creek plug: He couldn't account for the fact that the error reporting via FogBugz worked perfectly and to the same SQL Server as the rest of the data.
A couple, timeout-free months later, my manager boasted that he had fixed the timeouts ("Look, no timeouts!"). I had to hold back from grabbing a rock and saying "Look, no tigers!" but I did ask how he knew they would have occurred to which I got no response. The timeouts did return (and in greater numbers) a couple months later.
I'm pretty content with how I handled the situation but I'm curious how the SO crowd would have responded to letting a superior/colleague implement a solution you know (or are very sure) is wrong and will likely waste thousands of dollars?