The trust's editorial standards committee has ruled that a show broadcast on 3 October last year looking at the issue of nuclear power and the impact of radiation gave a "misleading impression" by failing to include research suggesting there could eventually be up to 16,000 premature deaths from the explosion and fire at the Chernobyl plant in Ukraine.
The trust ruled Bang Goes the Theory failed audiences by not looking at the wider impact of radiation, particularly given its mission to reveal "the truth about the effects of radiation".
"Viewers would be likely to be left with the impression that a relatively small number of deaths was the only serious adverse health outcome from the radiation fallout from Chernobyl," said the BBC Trust.
"The committee considered this would be a misleading impression based on the evidence and there had been a breach of accuracy in respect of how the programme reflected the health effects of radiation fallout from Chernobyl."
Bang Goes the Theory was looking at the issue of nuclear power in the wake of the reactor meltdowns at the Fukushima plant in Japan following the March 2011 earthquake and tsunami.
One section of the magazine-style show aimed to "wipe the slate clean and find out the truth of the effects of radiation".
The show argued that in the wake of the bombing of Hiroshima and Nagasaki during the second world war an inaccurate public perception of the actual health risks from nuclear power plant disasters such as Chernobyl and Fukushima had developed.
The BBC Trust's editorial standards committee (ESC) received a complaint on behalf of more than 50 co-signatories that the show was "extremely selective" in the figures it quoted about the impact of radiation released following the Chernobyl disaster and minimised the "more significant and contentious issue" of the secondary effects of health problems such as thyroid cancer.
Bang Goes the Theory cited the Chernobyl Forum report as authoritative and one of its key sources for its nuclear power piece.
But the ESC found that the show failed to mention in its broadcast comments made to the BBC in March last year by one of the report's principal authors that there is an expectation of 40,000 extra cancers as a result of Chernobyl, resulting in 16,000 premature deaths.
A range of other evidence assessed by the ESC included a report showing there have been 6,000 cases of thyroid cancer reported in children and adolescents as of 2005 – with many more expected over the coming decades.