Britain's Colin Oates missed out on a medal in the -66kg category, as Lasha Shavdatuashvili claimed gold.
Oates, the only British judoka to progress beyond the first round so far, lost his repechage bout to South Korea's Jun-Ho Cho by two yukos.
The Briton, who finished equal seventh, caused an upset by beating world number two Tsagaanbaatar Khashbaatar in the third round.
Georgian Shavdatuashvili beat Hungary's Miklos Ungvari in the final.
The bronze medals were won by Masashi Ebinuma of Japan and world number eight Cho of South Korea.
GB judo at the Olympics
- Britain's last Olympic judo medal was the silver won by Kate Howey in the -70kg category at Sydney in 2000
- The last British man to win an Olympic judo medal was Ray Stevens, who took silver in the half heavyweight category at Barcelona in 1992
Earlier, in front of an electric crowd at the ExCel Arena, the 29-year-old Oates raised spirits in the British team by beating Australian Ivo dos Santos in his first bout, having received a bye into the second round.
And he then sent the home fans into raptures by beating Mongolia's Khashbaatar, an Olympic bronze medallist in 2004. Oates scored the winning yuko knockdown with eight seconds left of the five-minute contest.
Norfolk's Oates then lost his quarter-final against eventual gold medallist Shavdatuashvili on a golden score, the Georgian claiming victory with a yuko score two minutes into extra time.
Oates still had a chance of a bronze medal via the repechage but Cho - who suffered a hugely controversial quarter-final defeat on judges' flags, having initially been given victory against Japan's Ebinuma - won by two yukos.
Oates had raised hopes of a first British judo medal at an Olympics since 2000 and a first from a British male since Ray Stevens won silver in Barcelona in 1992.
"I am disappointed because I got to the stage where I had given myself a chance to fight for a medal," he said.
"Today was one of the best bouts I have ever fought, especially against the Mongolian, who has been so consistent.
"It was a really good win to get past him, but I felt the Georgian was my opportunity, because the Korean was always going to be very tough.
"I went for a big score, but it did not work out."
lost her opening -52kg bout to Ae An Kum of North Korea.
Cox, who returned to the sport in 2010 having retired following the 2004 Olympics, battled well during the opening stages.
However, the 2008 Olympic silver medallist, caught Cox with a yuko score with one minute and 45 seconds remaining, which proved enough for victory.