Those 60's Browns teams certainly had their peaks and valleys, especially after 1964. 1967 was the worst, losing to Green Bay 55 to 7, and to Dallas in the playoff game, 52 to 14. As humiliating as the Green Bay loss was, the Dallas playoff loss was even worse because it was a playoff game, and Cleveland had 7 turnovers in the Green Bay game and two kickoff returns for TD's (Travis Williams). Dallas just plain outplayed the Browns in just about every way possible.
Cleveland did not play very well on the road that year, and both the Green Bay game and the Dallas playoff game were on the tail end of 3 game road trips. They also lost the 1957 championship game, 59 to 14, on the end of a 3 game road trip. Also the 10 to 0 loss to the Giants in the 1958 playoff game and the 1951 championship game loss to the Rams, 24 to 17, were on the end of 3 game road trips. I guess that the best time to play the Cleveland Browns in a big game was to play them when it was their 3rd straight game on the road.
As for 1967, Cleveland only lost one game at home, and it was in the opener against the Cowboys, 21 to 14, as Dallas completely stymied the Browns running game. Still, Cleveland missed out on 3 field goals that game and they were not from far out. Groza, in his last season missed 2 of them and holder Shiner flubbed the pass from center on another. It was maybe the best game Cleveland played all year.
Then there was the Detroit game 31 to 14 loss in a post Dallas game. Detroit seemed to run at will against the Browns that day, and Cleveland seemed to be clueless about how to stop them. The Lions made 29 first downs in that game.