The visual anomolies you saw are referred to as 'artifacts' and they can be a sign of a card dying. They definitely were with my older 7800 GS in my socket A system. It's in a very bad state. Still works for 2d apps, but when you boot, POST is garbled (strange characters and misspelled words), and 3d apps artifact and sometimes BSOD the computer.
As earthdog suggested, check for temp related problems. Also, shady power supplies can be at fault as well. An improperly seated cable or video card can cause minor and major problems, might check that as well.
When I replaced the broken card with a (broken) replacement the problems all went away, and the monitor was fine. Unfortunately the card I bought does indeed start showing artifacts, but only under high temps and heavy load.
So from experience: Yeah, it can be a sign of your card dying, but it doesn't mean that's what it has to be.