At least twice when President Kennedy and his cabinet are talking in a conference room, microphones from the ceiling hanging down can clearly be seen. At one point there are even two visible microphones hanging from the ceiling directly in the middle of the screen.
This Happened in almost every scene in the movie, at least in the theater version. I can personally think of a least 13 different instances of when this happened myself, and I'm sure that there are more. The editor was obviously not paying attention to the boom mics.