I think the "trust the authorities" thing was on its last legs by the sixties at least. Look at QUATERMASS AND THE PIT (aka FIVE MILLION YEARS TO EARTH). In that case, the military authorities are actively obstructionist, refuse to listen to Quatermass and his fellow scientists, and actually end up making things worse. It's up to Quatermass and a colleague to keep us all from being turned into Martian drones!
Of course, that's an English movie. I can't immediately think of an American example from that era.