I never had any issues with PC PSUs until AMD Athlon/Duron era. At this point the local market was flooded with cheap, nasty Chinese inventions that had ever higher W/A numbers but the insides were more and more "optimized" for mass production. I actually had 2 or 3 go bang on me in the summer, and at least one died when the PC was doing typical office work. It was at this point that I've started to be very particular about PSU brands. Before that I would only replace the fans and even that was rare, I didn't even bother with capacitors (nothing wrong with them visually and I didn't have tools to properly test them otherwise).
Years later I got some very tired AT PSUs that required cap replacement (not to mention cleaning), even some with blown primary as the results - I've fixed all of those that had at least decent build quality (even the no-names) and so far had zero problems afterwards.
I assume these issues would also affect Pentium III, I never had one. I do know of a few P4 systems that suffered from dried out caps in the PSU, but then again that CPU was a serious power hog and many people talked into buying one didn't know better, so the PC they've built (or bought from small local shops) were not cooled properly, that certainly didn't help.
Tl;DR: PSU quality went downhill, power usage went up a lot. PSUs went bang. Before that an AT PC with 486 and one HDD could be easily powered by 150W PSU with quite a safety margin too.