Earlier this week, I powered on my PC and the memory test failed. After running a few of the usual tests (reseat the DIMMs in their sockets, try each DIMM individually, try different sockets) I determined that one of my DIMMs no longer works reliably. (This DIMM was working fine for two years until this weekend.) I removed it and sent it off to the manufacturer for warranty replacement.
If I hadn't enabled memory testing, leaving it in the "fast boot" mode, I would not have discovered this problem. Windows and my applications would simply run, and then malfunction when accessing the defective parts of RAM. I'd have then started tracking down things like buggy software and/or virusses, when the problem was a piece of hardware gone bad.
To everybody reading this: learn from my experience. If your PC is configured to skip memory testing (sometimes referred to as "fast boot" or "quick boot"), change that configuration. It may make you wait an extra minute or two when turning the computer on, but you will be able to quickly detect memory problems before they cause your software to malfunction.