In June 8, 1967, on the fourth day of the six day war, Israeli airplanes and boats attacked an US spy ship called the USS Liberty. 34 US soldiers were killed, but the attack was aborted once the ship was identified as a US ship. The ship was close to the Sinai desert that was in the process of being occupied by the Israeli forces.
The Israeli version, is that the ship was unidentified, did not carry a flag, it was in a war zone, and was falsely identified as an Egyptian ship. American stated (in the UN) that all their ships are at least 300 miles away from the war zone. As soon as the ship identified itself as an American ship the attack was aborted. This version is problematic, since it seems like the ship was identified the previous days, and somehow this identification was lost. The ship was twice bigger than the Egyptian ship it was identified as, and the ship was covered with Antennas – not exactly a threatening presence. The numerous Israelis involved in the incident swear by their not yet dead grandmother that the ship was NOT showing any flag. In any case, even if we accept the Israeli version, is indicates a scale of incompetence previously demonstrated only by the guards of Troy that decided not to check the horse before dragging it into their city.
The American version is also problematic. The ship received direct orders to leave the theater the day before. Strangely the ship failed to receive the commands (but it received messages before and after this message without any problems). The US sailors claim that the American flag was flying high, but the ship was moving too slowly for the flag to be flying high (and the Israelies claimed there was no flag).
So far, a story of incompetence and stupidity, and at least one side seems to be lying a bit (I believe the Israeli version, you might prefer the American version). But there are numerous conspiracy theories about why would Israel knowingly attack an American ship. Most of these theories do not make any sense, but they seemed to be fueled by the inability to believe that the IDF (Israeli defence forces) could have been that incompetent (while winning the six days war so convincingly).
The Israeli navy at the time was small, underfunded and of low priority for the Israeli army. Not to mention that the Israeli ships were WW II scrap quality stuff. It seems that this level of incompetence is quite believable. But the conspiracy theories, once out of the bottle, will live on.
See here for many more details (than you want to know) about this story and numerous conspiracy theories and their limitations.
Just remember – any part of a conspiracy theory that can not be explained more convincingly by incompetence, can be explained by higher level of incompetence.