I do with 3.4GHz Pentium 4 and old 8800GT, especially HD ones. Also have problems with mp4, mkv and similar, only XVID/DVIX are ok.
So if you have that PC already, do not waste money for upgrades, if you don't, do not get it, for HD streaming you need at least Dual Core of any kind.
Streaming depends far more on your ISP than on your system. Since there's almost zero computing power required a Pentium 4 is far more than enough. You do need ~2GB of RAM and a GPU that supports 1080p, but pretty much anything from the last decade does that. Older integrated graphics may not be good enough, though.