The Warren Commission fingered Oswald. And as recently as the summer of '63 Oswald was spending his time protesting in the "Hands off Cuba" movement.
If Castro did kill Kennedy, then the government 1) wouldn't want to admit a "defeat" to a communist. And 2) wouldn't want to incite the American people into a frenzy to invade Cuba, and probably the USSR as well. Had we admitted that Cuba killed Kennedy, nuclear WWIII would probably have started within the week.
Brokeback Grassy Knoll?
I believe the story is that Johnson covered up the Castro connection because he didn't want to deal with the reaction of the Amercian public to Kennedy's assassination so close after the missile crisis.
Plus Johnson was afraid it could push voters into the Republican camp too close before the 1964 election.
I have no idea what the evidence is for this story or whether it stands up to scrutiny.