Artifacts on only 1 of 2 monitors

(I'm putting this in this forum because I'm fairly certain it's a monitor and not a graphics card issue)

I have a dual monitor setup. One of them is a 19" 4:3 Eizo M1950 that is 5-6 years old. The other one is a 32" LG HDTV whose exact model currently escapes me. Recently I've started getting artifacts on the old monitor. They only appear on that one, never on the TV, and they cover the entire screen. Also when they disappear, they disappear completely without a trace. There's never a middle ground, it's either the entire thing, or nothing, as if a filter or something is applied to the image. Sometimes they come and go at a flickering-like pace. The monitor's temperature or running time seems to be irrelevant, as I can go for entire days without them showing up, only to get them again next morning 10 minutes after I turn the PC on. My GeForce 460 GTX is almost always idling at ~43-45C when it happens.

What gives? Is it the monitor? The DVI cable? Is it even possible for one of the 2 DVI ports on the card to go "bad"?

What happens if you swap the monitors on the 460 GTX outputs?
Are the artifacts still on the same monitor?

They still only appear on the one monitor, so the outputs don't seem to be at fault. What I haven't done yet, is switch to a different DVI cable altogether, seeing as I don't have one (the TV is connected through DVI-to-HDMI). I guess I should see if I can find one to check and confirm.