Unless you're one of those people who have the worst luck in the world (ie you've been electrocuted by your refrigerator or have lit yourself on fire --accidentally-- and received severe burns), running a failing card shouldn't
hurt anything. There's a remote chance, but it's very remote.
It's one of three things, and yes, the card failing is most likely. It's also probably the one you want it to be as it's the easiest to find. The other two options are PSU failure and mobo failure.
First thing to do is try to get access to a friend's computer (or your wife's if hers has the parts for it). Try your card in their computer see if it does the same thing. If it runs fine on their computer, it gets a little more difficult. If it still fails in their computer, cards gone, get yourself an upgrade. I'd recommend a 570 over 2x 460s though.* Try their card in your PC, see if there are any problems. Then there there's a couple different questions:
Is their card more powerful in a power consumption way (look around online for this or give us the info to find out)?
If yes, nothing is solved.
If it's a more efficient/weaker card, and there are still problems, it's unlikely that it's the power supply, your motherboard might be on the outs.
If there are no problems with a weaker/efficient card, it might be your power supply.
Now, try putting your card back in your machine, and putting in their power supply. Just connect the card/mobo/main hard drive. Don't bother rewiring everything.
If there are no problems with this setup, your power supply is failing.
If there are problems with this setup, and there were problems with the other card as well, your motherboard's PCIe slot is failing.**
If there are problems with this setup that weren't there with the other card, swap in that card and try again with your mobo + test PSU.
*570 vs SLI 460: Performance gets a nod to 460s, but you've got no upgrade path with 460s. With a 570, you could always add one (or two) more down the road.
**I've had this happen before. I bought a new card and then returned that for a new one thinking it was DOA. When I had 3 different cards not usable, I finally looked elsewhere for a problem. I did neglect to go back to my other card though, it's sitting in a box on the other side of the room