Second time I've ever been beaten boat over boat

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Ultimate Bet No-Limit Hold'em Tourney, Big Blind is t60 (6 handed)

CO (t1650)
Button (t1640)
SB (t4760)
Hero (t2005)
UTG (t2950)
MP (t1995)

Preflop: Hero is BB with , .
UTG calls t60, 1 fold, CO calls t60, Button calls t60, 1 fold, Hero checks.

Flop: (t270) , , (4 players)
Hero checks, UTG checks, CO checks, Button checks.

Turn: (t270) (4 players)
Hero checks, UTG checks, CO checks, Button bets t270, Hero calls t270, UTG folds, CO folds.

River: (t810) (2 players)
Hero bets t1675 (All-In), Button calls t1310 (All-In).

Final Pot: t3795
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The first two checks were because the cutoff had been playing like an absolute idiot and I figured I could trap him.
Is there any way out of a hand like this? Could I have played it some other way that might have tipped me off to him having a bigger boat?

Like I said... if it gets to the river, you're fucked... I wouldn't fold a boat there if my mom and Jesus were both asking...

I think you have a chance to get away if you bet out the flop and turn, and get action back... I mean, what the hell could he be playing there??? If you bet out the flop and turn, you're representing that you have a 4... If he doesn't seem afraid, tuck tail and run...