After a 12th defeat in 13 away games, perhaps only a genuine humiliation will awaken them. On this evidence, it's certainly coming.
Striker David Nugent epitomised Burnley's poverty. Such was Fulham's midfield monopoly that he hardly got a touch, and spent the game alternately haring after lost causes and shaking his head sadly. Not that the home side were in top gear. There was no need. They simply waited to be presented with the ball and passed it around at will.
Fulham have often struggled with the favourites' tag. But with the visitors content to play the game inside their own half, they enjoyed all the early pressure and had time to pick the telling ball.
It was Nicky Shorey who provided it, floating a cross on to the head of David Elm. The Swede held off David Edgar and nodded the ball into the path of Danny Murphy, the midfield general making a rare sortie behind enemy lines. The captain made no mistake from six yards, although Elm may have been a touch offside.
It was a familiar but ever perplexing story for Burnley: a side so elegant and assured on their own patch looked like they were playing on ice. Two free-kicks, one fired straight at Mark Schwarzer by Danny Fox and one from Tyrone Mears that clipped the upright, were the limit of their potency.
With a dulling inevitability, they went two down. Bobby Zamora beat the offside trap and shot low from an angle. Brian Jensen blocked but should have held, and Elm was on hand to swipe home his first goal in English football.
In such promising circumstances, it was only a matter of time before Zamora ended his six-game league drought. He should have scored when a clearance hit him and fell at his feet 12 yards out, but made amends with a low left-footed free-kick from 20 yards.
Burnley were fortunate that it stayed at that, having created virtually nothing of their own. They now enjoy a break of almost a fortnight before playing Aston Villa, but this week could yet hold more ignominy: Rothes travel to second-bottom Fort William on Wednesday.