The crucial mistake came long before that. In his memoirs, and the derived book on Vietnam, Kissinger claims that Ike, on leaving office, urged Kennedy to defend Laos at all cost, claiming that it was the essential strategic front against the North Vietnamese.
Clearly Ike had it right, and Kennedy screwed the pooch with his myopic focus on Vietnam alone (as opposed to the strategic big picture in Indochina). Once the NV had established bases all along the borders in Laos and Cambodia (and had secured, however perverted, the precedence that any attempt to dislodge them was a violation of Laotian and Cambodian sovereignty) the game was pretty much up. The North Vietnamese could then maintain a guerrilla insurgency in South Vietnam indefinitely. Short of a massive commitment (hundreds of thousands) of troops, we were basically screwed from that point on.