Thank you PsychosisNode for your suggestions.
I followed your advice and cleaned the card. I also tried hitting the monitor a bit. In the end I was able to rule out everything but the graphics card. Eventually, things got so bad that the text during the boot process was barely readable because individual characters were replaced with other, random characters (clearly not something a bad monitor would do). And the computer would freeze when it got to the splash screen.
I went out and bought an ATI Radeon 9200 to replace the NVIDIA Geforce4 4200 ti I was using. Unfortunately, I couldn't get the ATI driver I downloaded for this card to install properly due to a library-file conflict, but the generic VESA driver I'm using now seems to work fine.
In other words, I'm back to normal operation. Thanks again