With the Challenger disaster, the contractor's engineering team that developed the booster gave a no-go for launch and NASA administrators showed off their engineering skills by using social engineering to trick an unqualified and unapproved employee of the contractor into signing the form to go for launch. That's far worse than being bad at calculating the safety risks; the specifically overruled the decision by the people that were good at calculating the safety risk.
Source: Truth, Lies, and O-Rings by Allan J. McDonald, James R. Hansen, and Jonathan Yen
Source: Truth, Lies, and O-Rings by Allan J. McDonald, James R. Hansen, and Jonathan Yen