Al-Lytic caps are probably highest failure rate part in all of electronics and any past 15 years old are on the right side of the bathtub curve.
Bad tantalum caps are showing up more and more often. Used to only come across that every few years. Now it's several times a year.
Bad RAM on a video card isn't that uncommon.
Inspect all the ICs (including MOSFETs) looking for cracks or pits or shiny melted spots.
Look for places where scratches have broken traces or any other physical damage.
Look for cold solder joints.
Try replacing the thermal paste.
For AGP make sure the card is compatible with the slot.
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GRUMPY OLD FART - On Hiatus, sort'a
Mann-Made Global Warming. - We should be more concerned about the Intellectual Climate.
You can teach a man to fish and feed him for life, but if he can't handle sushi you must also teach him to cook.