World champions Nathan Cohen and Joseph Sullivan won the Double Sculls in 6:31.67, ahead of Italy and the Slovenian crew who were looking to reclaim the Olympic title they won 12 years ago in Sydney.
Great Britain's pair made a promising start, leading the field after 250m and were in medal contention heading into the second half of the race.
But New Zealand and Italy produced strong finishes to overhaul Slovenia and row Britain out of the medals.
South Africa won gold in the men's Lightweight Four as Great Britain won a silver medal after being pipped to the line in a sensational final.
In a thrilling sprint for the line, the British crew edged 2008 champions Denmark into bronze.
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