the outdoor storepsychiatric expert witnessime Cavendish's group missed by only 2:33.
"It's disappointing. We thought we were way out of the time limit with quite a long way to go but it was a lot closer and had we known it might have made a difference," said Cavendish.
Andy Schleck's hopes of securing the yellow jersey awarded to the race's overall leader last year were dented when eventual winner Alberto Contador exploited his jammed chain on the 15th stage.
But Schleck took revenge in emphatic style as he surged through the Tour's highest-ever finish 3:49 ahead of the demoralised Spaniard.
Four minutes 44 seconds, an improbable margin to recover over the three remaining stages, now separate Contador from a successful defence.
"Victory is impossible now. I had a bad day," said the Spaniard.
"My legs did not respond and I suffered an incredible collapse. It was a very hard day from the start."
Schleck attacked on the way up the Col d'Izoard, with more than 60km left to the finish, and quickly put two minutes into the peloton, with no immediate response from Contador, Evans or Voeckler.
With older brother Frank marking the other main contenders, Schleck received help from Leopard-Trek team-mates Joost Posthuma and Maxime Monfort.
The huge stakes involved in his gamble became clear as he established himself as the yellow jersey 'on the road', but facing a draining 20km climb to the safety of the finish.
At one point he increased his lead beyond four minutes but Evans led a fightback on the climb up the Galibier.