Check with Nikon, but as of a year ago they would replace the shutter on any D600 regardless of whether or not you were the original owner or not. I bought a D600 last summer specifically because it hadn't been replaced. I had another that had been back and forth a couple times, and was eventually replaced by a D610. I got the second one used, with grip and 18-105mm for $900, and immediately sent it to Nikon for the shutter work, figuring for that price I'm getting essentially a fully refurbished D600 - a steal!!
2000 shots later I was experiencing dust issues (not sure if it was a bad shutter or just dirt - I suspect the latter), so I contacted Nikon, they told me to send it back and they sent me a D610 to replace it.
$700 for a refurb D600 with zero clicks on a new shutter is a great buy for a great camera. Again, not sure if they're still doing it, but if they are I'd buy with far more confidence than if they're not. This is the announcement, so at worst if they replace the shutter and you have issues they're replace it with another new D600.
http://www.nikon.com/news/2014/0328_01.htm
I'll say this, it'll give you 2-3 stops of noise improvement over the D7000 (what I moved up from). Great camera. D610 is better, only in that it has a couple new features (Qc mode) and better body sealing. If you're lucky, you could get one of those.
