Paul Goodison survived disqualificiation in his final race in the Laser class - but still saw his hopes of a medal disappear.
The 26-year-old from Sheffield needed to finish in the top 19 and ahead of Slovenia's Vasilij Zbogar after the initial race was abandoned.
He came in 17th but Zbogar was 13th to claim the bronze, with Goodison fourth.
Brazilian Robert Scheidt, pipped by Ben Ainslie four years ago, took gold with Austrian Andreas Geritzer in silver.
Goodison, a point behind Zbogar before racing began, slipped around 15 seconds behind the Slovenian as they hit the second mark.
But a strong down-wind leg saw Goodison tack in front of Zbogar as they rounded the third mark to raise his hopes.
But he reached the fourth mark four places behind the Slovenian, and was unable to make up the deficit on the final down-wind leg.
He ultimately finished five points outside the medal positions.
Earlier, Goodison had failed to go back after being over the start line when racing belatedly began an hour late.
But as the wind dropped 15 minutes into the race and it was abandoned, his disqualification did not count when a new race began just 15 minutes before the final cut-off point of 1600 local time (1400 BST).
Goodison was bidding for Britain's fourth sailing medal of the Games after the golds won by Ainslie in the Finn and Shirley Robertson's Yngling crew, and the silver collected by 470 duo Nick Rogers and Joe Glanfield.
That may now come from Chris Draper and Simon Hiscocks, who moved into third position in the 49er class on Sunday.
Starting the day in fourth place overall, the European champions followed an eighth place in race 10 with a second in race 11 of the 16-race series.
They are now six points behind joint leaders Spain and Ukraine, and six clear of Australia in fourth.
Iain Percy - a gold medallist in the Finn four years ago - resumed his campaign in the Star (Keelboat) class with Steve Mitchell on Sunday.
The British duo finished 12th in race three and ninth in race four to drop to eighth place overall, 11 points adrift of the medal positions.
Leigh McMillan and Mark Bulkeley dropped to 15th overall in the Tornado (Multihull) class, after a 17th place in race three and a 12th in race four.
Laura Baldwin completed her campaign in the women's Europe class with a fourth place but finished 23rd out of 25 overall.
The gold went to Norway's Siren Sundby, with Lenka Smidova snatching silver for the Czech Republic and Denmark's Signe Livbjerg winning bronze.