Retronaut and other sources (including Connery's own website) claim he finished third in the Mr. Universe competition in 1953. But when you look at the group photo, that seems unlikely: He's not big enough.
His Wikipedia pages also refutes the claim, saying he either finished third in the Junior class, or failed to place in the Tall Man classification (he's 6 feet 2 inches).
Still, he looked impressive oiled up.
