If you have done the legendary quest to get the ilvl 680 ring, you can buy a second one and the game will factor it into your average ilvl even though you can't actually equip the second ring. Not super useful, but it can be good on alts to boost their ilvl a little bit to meet the molten core and highmaul LFR queue requirements if you didnt hit it while running heroics for the ring quest.
Edit: I have only done this with two different rings (tank and dps rings as a paladin), not two copies of the same ring.