Except for in very extraordinary cases I am sorry to inform you that connected passed pawns on the 6th rank always defeat a rook. 
Two pawns on the sixth rank versus rook
"Two passed pawns on the sixth rank beat everything up to a royal flush" GM Ian Rogers (quoted from the last link)
They also beat a lone rook that can't be assisted by its King. Three connected passed pawns on the 5th-rank also can't be stopped by a lone rook. Yeah I see the study right above me. It's very nice. As soon as it or something similar occurs in a real game let me know about it right away. I'll be waiting right next to the phone for your call.
http://www.irlchess.com/2012/12/09/rook-v-three-pawns-part-i/
http://www.chess.com/blog/Black__Knight/rook-vs-3-pawns?_domain=old_blog_host&_parent=old_frontend_blog_view
http://www.chesstape.com/index.php/endgame/3-connected-pawns-vs-rook
http://www.chesscafe.com/text/skittles315.pdf*
OK NimzoRoy, phone number please. Here is Keres-Eliskases, Nordwijk 1938.
Remellion I'm impressed by your find, esp since it stars two world-class GMs. Eliskases may not have been in Keres class but he was definitely a world-class GM in his heyday. BUT I'm still not losing any sleep over something like that happening in one of my games. I'm still trying to learn how to beat a K+R with a K+Q, which is slightly less obscure than this one.
I would give you my phone # but it's currently being tapped by the NSA and they're keeping track of everyone who calls. Instead I'll give you links to some (hopefully) interesting endgame blogs I've posted
http://www.chess.com/blog/NimzoRoy/3-minor-pieces-vs-2-rooks
http://www.chess.com/blog/NimzoRoy/a-famous-bishop-vs-knight-ending-or-smyslov-blows-botvinnik-off-the-board3
http://www.chess.com/blog/NimzoRoy/krrp-vs-kr
I found a pretty detailed article on K+R vs K+2 connected passed pawns:
http://www.chess-insights.com/rook-vs-two-connected-pawns.html
Except for in very extraordinary cases I am sorry to inform you that connected passed pawns on the 6th rank always defeat a rook.
Literally thought I was winning with two p's on 6th against R yesterday and lost the game as a result (it was actually a draw with best play, not winning at all).
Hello!I've been playing against this somewhat intermediate computer(it gives you a free rook or minor piece every once in awhile).Around 45% of my losses against the computer are because of 2 pawns on 6th rank defeat my extra rook very often.What are some tips against this endgame?