"one or more fans have failed."
if your talking to me then the reply button makes conversation go easier.
also i know that it said that but its obviously wrong if the fans turn. if it is the cpu fan that is spinning on start up then damage would be hard to do in the time it takes to start an intensive program and to see if it spins faster.
i once got a message the my gpu fail, rebooted and the gpu is fine. yeah the gpu got hot or something and failed shut down but that doesnt mean that it was broken it just meant that i needed to back off my overclock.
to the OP, i think more info is needed in order to figure out whats wrong. when you say "nudges slightly back and forth" do you mean the blades move but just a little like its stuck? also when you say you start up the pc and the fan spins "once" are you saying it spins one revolution or spins up fast then stops and never spins up again?