Cassidy and his two co-accused - 54-year-old Eugene McLoone from Abercorn Road and 46-year-old Danny Doyle from Quarry Street - are all charged with possessing an improvised explosive device with intent to endanger life.
Doyle previously served a prison sentence in the 1990s after arms and explosives were found in his house.
The trio appeared at Bishop Street Courthouse on Thursday amid tight security.
A detective told the court that Cassidy claimed he was the one to find the bomb and that he was worried children may find and play with it - so he was taking it to a priest.
He claims he was given a lift by McLoone, who drove past in his car accompanied by Doyle.
A defence solicitor said that, according to Cassidy, the other men didn't even know he had the bomb with him.
But District Judge Barney McElholm said the defendants' version of events completely lacked credibility and he refused bail.
As the defendants were led from the dock, members of the public stood up and applauded.