One rule is if the time traveler went back in time and saved John F. Kennedy, he would not have a reason for going back in time in the first place, so lets say he stops Lee Harvey Oswald, he still needs a reason to be there, why did he make the trip in the first place in Kennedy was never assassinated, so the timeline steps in to give him a reason to go back in time and safe JFK, so somebody else kill JFK. What if this time it is a KGB agent who confesses that he was sent by the Soviet Union to assassinate Kennedy in revenge for the Cuban Missile Crises, and this causes LBJ to declare war on the Soviet Union for this outrageous act. The United States wins this war, as the Soviets lack long range missiles, a few missiles that the Soviets do have get through and destroy American cities, Soviet bombers are mostly shot down, but the Soviet Union is devastated by the American response, Europe is devastated as well because the Soviets use their short range missiles so there is a USA to send a time traveler back to save JFK and to prevent World War III as well. World War III came as an unintended consequence of trying to save JFK, JFK is not saved, but World War III is added as an additional reason to go back in time to prevent.
The time traveler doesn't remember this change in history. the consequences for temporarily saving JFK's life aren't immediately apparent, he enters his time machine thinking that the mission has been accomplished, he still has the original timeline in his records and memory, he goes back to the present to find a world that is recovering from a nuclear war. Conservation of mass requires that the time traveler needs a reason to go back in time, and that the main reason for going back in time can never be fulfilled, but unintended changes to history can still occur during the mission, just so long as they are not the main reason for going back in time.
What do you think would happen if someone went back in time to kill Hitler before he starts World War II? Maybe someone else starts World War II instead, maybe by Joseph Stalin. A Soviet started World War II would probably occur in the 1950s. Without the Germans starting World War II, the United States doesn't perceive a threat, and does not build an atomic bomb, the jet engine is developed however and so is the Helicopter. FDR lives a bit longer without World War II, he retires after his second term in office.
The Soviet Union invades Europe, and it is only the fact that most European countries have disarmed and that the Great Depression took a great toll that the Soviets have a chance. The Soviets get stalled for a time at France's Maginot Wall, it takes them a while to figure out that they can go around it.