I had/have a K7 500 (early one, so the chip is actually marked 500MHz, think it will go to 550 if I push the voltage up a bit) on an Abit KA7 motherboard (originally on a MSI 6167 board that I don't remember at all). That had most of the big caps (>1000uF) go bad around 2003ish, with the rubber seals blowing out. I'm not sure how to diagnose the problem properly if there's no obvious visible faults, probably get an oscilloscope and measure the ripple voltage, and remove a few of the capacitors to check their capacitance and ESR with a component tester.
Symptoms at the time were gradually increasing instability, until eventually it wouldn't boot. When I finally had a look I also found that the PCB under on of the power FETs had darkened from getting too hot. At the time I changed out the all the capacitors that look obviously broken, and any others of the same value, and it's been fine since. The only major snags I had were unsoldering from the ground plane with the gas iron I had then, and trying to find capacitors with the same spec that would fit in the space. There weren't any, so the diameter of the replacements was a bit too big, so the placement got a bit, er, creative.
I don't remember the heatsink on the CPU getting all that hot, but that would have been helped by the heat transfer goop they used on the cache chips not actually connecting the chips to the heatsink (discovered when I finally cracked open the CPU case).
Datadrainer, AIUI, the caps will have a rated voltage, temperature and lifespan. Often the lifespan at their limits isn't very long (couple of thousand hours). The caps are meant to reduce ripple in the voltage by shunting the ripple current to/from ground, and ideally there is no resistance to that ripple current, so there's no actual power burnt in the cap. But there is always some resistance. As the capacitor starts to fail then the resistance will increase, which means there will be more power burnt in the cap (so it gets hotter) and higher ripple in the voltage (so it'll run closer to it's voltage limit). Both of those things cause more current to flow through the other capacitors, which pushes those to fail faster as well. Eventually the ripple will get big enough that it starts causing glitches in the CPU, and probably causes the power FETs to switch more often, so they get hotter as well. So a slight increase in the resistance in a few caps can cause a load of escalating problems.
What are the capacitance, voltage and diameters of the caps near the CPU slot on this board? There must be some pretty close equivalents.