I bought a computer from HP with a Radeon HD 6570 graphics card. The fan on the card recently started making a horrible buzzing sound and as the computer is a few months out of warranty, I'll have to replace it. I went to the AMD site to buy a replacement and I noticed that a 400W PSU is recommended. I checked inside my machine and it came with a 300W max PSU. Could the weak PSU have caused the fan to be running too fast and worn it out?
No. If anything, an underpowered PSU would cause the fan to run slower because the card might drop into a reduced functionality/low power mode.
But what you describe sounds like a bearing going out. There's no real rhyme or reason to why bearings go bad when they do. Sometimes they go bad almost immediately, sometimes they go pretty much forever, sometimes you can even have bearings making a huge amount of noise that keep on chugging for months or years.