I shied away from upgrading the power supply -- which would have been a major hassle -- and sought other solutions. As it turned out, the software control suite for the video card (an ATI Radeon HD 4650) allowed the user to manually override the GPU and memory clock speeds, as well as the fan speed. I set all of these to the lowest possible settings (see the illustration), reattached the second display, and haven't had a problem since.
The Catalyst Control Center for ATI video cards lets you change a whole bevy of normally hidden settings.
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I should point out that other devices, such as hard drives and optical drives, typically don't draw all that much power. Removing them as a power-saving measure (as opposed to debugging, as described above) gives you back very little.
One common reason for random failure is issues with memory -- a bad memory module can appear even on the most high-end machines. The best way to determine if there's a memory issue with a given machine is to test it, rigorously and repeatedly. Vista has its own memory test application, but you can also download and run a program like Memtest86+ (http://www.memtest.org/), which sports a slightly broader set of test parameters.
The best way to run a memory test is to set it up and let it run overnight: not just one pass, but continuously, for hours on end. If the test program detects an error -- or, worse, if the machine locks up solid -- there's a good chance one of the DIMMs is defective. Sometimes mismatched DIMMs can cause problems; try pulling one and then the other, and see if things go south on you then.
Memtest86+ puts your system's memory through a grueling battery of workouts.
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Get Everything Up-To-Date
This means more than running Windows Update. Your PC manufacturer may have updates not offered through Microsoft -- BIOS patches, for instance, or device drivers not provided in the default Windows installation. Fortunately it's become that much easier to find these things and keep them current -- Sony and Dell, for instance, both have applications that bring you directly to the relevant web page for your system. BIOS updates often go by undetected, both because they're generally not delivered automatically and because many people are still twitchy about applying BIOS updates. They shouldn't be: in the past, updating BIOS typically required booting a DOS disk or something similar, but today the vast majority of such updates can be done from within Windows, quite safely.
The single most important thing -- and the one hardest to remember for many people -- is to be patient and diligent. It's easy to succumb to the temptation to pitch the whole thing out the window and start anew, but that's also an expensive solution -- and brings with it the risk that you'll end up no better off than you were before. Solve a problem like this on your own (or with a little guru oversight), and you'll be that much better equipped to tackle something like this the next time it shows up.
InformationWeek has published an in-depth report on Windows 7. Download the report here (registration required).