Not in all cases. The H60 from Corsair is a very basic model and will likely get rather loud trying to keep your CPU cool, assuming it doesn't just overheat.
Also, it's very important that you don't skimp on the PSU (which you have). The PSU delivers voltage to all of the internal components, meaning that a low quality PSU can reduce the lifespan of your components. Also, if the PSU were to fail, it could destroy other components as well.