The Euxton pole vaulter was in action at a packed Olympic Stadium.
The Great Britain number one took on some stern opposition, as Katerina Stefanidi closed with a height of 4.80m to win, from Yarisley Silva and Eliza McCartney.
Bradshaw, 24, now based in Cardiff, was fourth thanks to a first time clearance at 4.52m, but was disappointed not to clear 4.62m.
She said: “It wasn’t the best performance from me.
“I don’t really know what happened, because the conditions were really good.
“I wanted to try and make the most of the opportunity.
“It is disappointing because it is my last competition before Rio.
“I’m still in a better place than I was going into the Beijing World Championships, so I have got to take the positives and work on a few things.”
The Olympics begin on August 5th and run until the 21st, with qualifying for the women’s pole vault on the 16th, and the final on the 19th. Bradshaw, currently ranked 13th in the world, was sixth at the last Olympic Games – her first – in London, after qualifying fourth from group A.
The same year she was a bronze medalist at the IAAF World Indoor Championships, going on to win gold in 2013 at the European Indoor Championships.
The 2011 European U23 Championships winner and 2010 World Junior Championships bronze medalist was seventh in Beijing at the Worlds last year, after winning qualifying group B.
Her best performance this year is the 4.65m to finish fourth at the Diamond League meeting in Monaco earlier this month.
The qualifying standard for Rio was 4.50m, and Bradshaw rubber-stamped her place on the plane by winning the British Championships in Birmingham last month with 4.60m.She was named in the Team GB squad recently, and was delighted to get the nod.
She said: “To be selected for my second Olympics is such an honour.
“What kept me going through my injuries was the thought of me making my second Games and to finally being going is a dream
“I have worked so hard and rounded into great shape over the past month.
“I just need a bit of luck with the weather in my next few competitions and hopefully I can do myself justice.”