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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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The 23-year-old has been linked with a move away from the Rec after his father Mike was sacked as head coach last May.
Ford is under contract at Bath until 2018, but is understood to have a break clause in that deal for this summer and Sale are keen to talk to him.
"If we get an opportunity we will," Diamond told BBC Sport.
"He's still in contract with Bath so we don't know where that is up to."
Asked if their current 10th-placed position in the Premiership could affect any offer to Oldham-born Ford, Diamond responded: "No, as people come for money and ambition, and if you've got the money then you've got the ambition - Wasps are a good example."
Sale signed USA international AJ MacGinty and Dan Mugford from Nottingham this season after Danny Cipriani moved to Wasps and George's older brother Joe left for Yorkshire Carnegie.
However, outside centre Sam James currently is playing at 10 for Sale who are on a run of nine defeats in a row in all competitions.
Meanwhile, Castleford Tigers have claimed they should receive £500,000 in compensation after Sale signed winger Denny Solomona, who has scored four tries in four games since his rugby union debut.
Solomona was under contract with the Super League outfit until 2018 but left them to sign a three-year deal at Sale.
Cas have launched legal action against the Sharks, Solomona and his agent Andy Clark.
"They openly said we'd made them an offer which was substantially more than what Warrington Wolves had offered them and they rejected that," added Diamond.
"We feel vindicated by what we have done as we signed a player after he resigned and they'd sacked him.
"Castleford's claim is 10 times what he was earning. Maybe they should have thought of that and paid him what he is worth one would suggest."
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.
[NEXT_CONCEPT]
Sale have yet to speak to England and Bath fly-half George Ford over a potential move, says Sharks director of rugby Steve Diamond.