Britain's Chris Draper and Simon Hiscocks won bronze in a tense final 49er race on the Saronic Gulf.
The pair were guaranteed a medal but needed to win the last race and hope Spain finished worse than 12th to have any chance of gold.
Draper and Hiscocks finished sixth and had to settle for bronze behind silver medallists Ukraine and champions Spain.
But in the Tornado and Star classes, neither of the British crews can win a medal with one race each left.
The Spanish 49er team of Iker Martinez and Xavier
Fernandez enjoyed a nine-point lead overnight and secured gold with a seventh place in race 11, while Ukraine stayed ahead of GB after a third.
Spain finished on 67 points with Ukraine on 72 and Great Britain on 77.
"It kind of feels like we were beaten today because we had the bronze and were trying to better it," said Hiscocks, who won silver with helm Ian Barker in Sydney four years ago.
"But we appreciate that we've come away with a medal which is kind of what the Games is about. We're pretty happy with it."
Draper added: "We were pretty upset when we finished. We came here to win gold and it's a bit disappointing. But it's a fantastic achievement really."
Britain's Star sailors Iain Percy and Steve Mitchell are seventh overall after a seventh and 17th place in races nine and 10 on Thursday.
The former Star world champions are on 69 points, 20 points adrift of third place.
Percy said: "I'm gutted, but I never
thought until after that race we wouldn't get a medal. I always believed we would, so that makes it doubly hard.
"I've had some tough days in the last year but that takes the biscuit.
"We didn't sail our best, but you have to fight back, not lose it all and
keep going. But it takes a big chunk out of you."
Brazilians Torben Grael and Marcelo Ferreira secured gold after an 11th and fourth. It was a record fifth Olympic medal in six Games for 44-year-old Grael.
France are in second place with Canada in third with both the silver and bronze medals up for grabs in the final race on Saturday.
In the Tornado catamaran class, Britons Leigh McMillan and Mark Bulkeley placed 16th in race nine and 15th in race 10 and are 13th overall with one race left.
Britain have now won five sailing medals in Athens to match their performance in Sydney.
Ben Ainslie won gold in the Finn and Shirley Robertson's Yngling crew also triumphed, while Nick Rogers and Jo Glanfield won silver in the 470s and windsurfer Nick Dempsey snatched a bronze in his final race.