Bigwood, Charlotte Dujardin, Carl Hester and Spencer Wilton finished second behind Germany in the team event.
Dujardin recorded the second best score of the Grand Prix Special, 82.983%, aboard Valegro but Isabell Werth ensured Germany stripped Team GB of their Olympic title.
Double Olympic champion Dujardin along with Hester, Wilton and Bigwood combined to produce an average score of 78.595 with Germany winning with 81.936.
Dujardin will ride for individual gold on Monday in the Grand Prix Freestyle and saluted Valegro with Team GB’s overall medal tally at the Games now standing at 20.
Team GB Rio 2016 medal tally: Gold: 6. Silver: 8. Bronze: 6. Total: 20.
Charlotte Dujardin said: “I haven’t ridden [the Grand Prix] test since the European Championships last year. I probably should have practised it a bit more but [Valegro] is such a genius.
“The last two rounds have been fantastic and it’s a huge privilege to be riding alongside these guys. We have new music for the freestyle, which will hopefully go to plan. I just want to go out there and enjoy myself and give Valegro a good time.”
Carl Hester said: “The Olympics produces a lot of nerves, a different atmosphere and different rides from people. I felt the hold of the team has ridden beautifully.
“Everybody coped with the pressure and the fact that these horses have travelled halfway around the world and come out and done performances like that is really testament to our team of grooms.”
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