The amphibious landing, on the Puerto Rican Island of Vieques, began on October 15th - the day experts at the National Photographic Interpretation Center (NPIC) were examining Major Heyser's U-2 photographs and first noticed a truck convoy approaching a deployment of Soviet missiles near Los Palacios at San Cristobal. The exercise was code-named "PHIBRIGLEX-62" and involved more than 40 ships, approximately 20,000 naval personnel and 4,000 marines. (Scroll down 25% to read about the exercise in this document located at the official U.S. State Department website.)
At the time, some folks thought the mock invasion was a pretext for another military exercise: the potential blockade of Cuba. Those who thought there was something beyond a mock invasion may have been close to the mark. By August 23, 1962, the White House had issued "Top Secret & Sensitive" (reproduction "prohibited") National Security Action Memorandum 181. Located at the JFK Library, the document is the first Kennedy Administration memo on the Crisis. It states (among other things):
A study should be made of the advantages and disadvantages of action to liberate Cuba by blockade or invasion or other action beyond MONGOOSE B...
...probable military, political and psychological impact of the establishment in Cuba of...surface-to-surface missiles which could reach the U.S.
The President himself dealt with early leaks to the press. Quoted in The New York Times, JFK chastised Senator Kenneth Keating (R-NY) for publicly speaking in the United States Senate regarding the presence of dangerous Soviet missiles in Cuba. Although the President called Keating "a nut case," it appears the Senator, in fact, knew about the presence of Soviet missiles in Cuba before the President knew. (On the other hand, JFK may have known more himself. A declassified U-2 reconnaissance photograph taken on August 29, 1962 reveals the construction of a surface-to-air [SAM] missile site at La Coloma while an October 5, 1962 CIA chart of "Reconnaissance Objectives in Cuba" lays out "unidentified - missiles.")
Even when the President had more reliable information himself, it was not yet time to inform the American people of events in Cuba. Not enough specific facts were known. The President needed more information and he needed to confer with his advisers - especially his closest adviser, the Attorney General Robert F. Kennedy.