Well I've never heard that exactly, only that the more your HDD fills up, the slower it can go. Defragging helps relieve this. My biggest hard drive has 12GB left and I haven't noticed much of a slowdown yet so I don't think it is too important to leave 20GB on your HDD.
It slowed my computer by a lot. I have about 12 GB left, used to be less. Now I can't even run Guild Wars because of the lag every 2 seconds.
I'll be getting an external to save some stuff so hopefully when I reformat it, it'll work smoothly.