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The top two were tied on a clearance of 4.85m, but European champion Stefanidi won by having fewer failed jumps.
Bradshaw, 24, set a season's best of 4.70m, with New Zealand's Eliza McCartney taking bronze with 4.80m.
Cuba's world champion Yarisley Silva managed only joint seventh with 4.60m, 31cm short of her personal best.
"I am fifth at the Olympic Games and I am really happy with that. It was a step up from the last Olympics I was at in London," said Bradshaw, who was joint sixth four years ago with 4.45m.
"I am an ultimate competitor so I am always going to be disappointed whatever happens. I felt I jumped the best of my season and I was so close to 4.80m.
"I am just a little bit too inconsistent to mix it with the top girls in the world at the moment. I just need a year where I don't have an injury."
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Defending champion Jennifer Suhr, the only athlete to clear five metres this season, was joint seventh and said she was suffering with an illness.
"I feel horrible,'' Suhr said.
"I'm actually getting concerned. It's going on day 10 and I feel worse today.
"They think it's respiratory and now in the lungs. This morning I was coughing up blood. It's getting scary.
"I just want to get out of here and go home and figure it out."
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Greece's Ekaterini Stefanidi won women's pole vault gold, narrowly beating USA's Sandi Morris into silver as Britain's Holly Bradshaw was fifth.