That policy allows Cubans who make it to United States shores to apply, ultimately, for asylum, but sends those picked up at sea back to Cuba.
As traffic jammed, Coast Guard officials relented, allowing the four other Cubans, who had been picked up in the water, to seek political asylum.
The sharp rise is inflaming tensions over a policy that allows Cubans without visas to stay if they reach American soil but turns back those caught at sea.
The policy allows Cubans who land in the United States by boat or other means to stay, but requires those intercepted at sea to return to Cuba.
In recent years, though, the Government has eased restrictions and allowed Cubans more freedom to worship.
The decision was accompanied by another - a good one allowing Cubans incarcerated on Guantanamo to enter America.
It also criticized US policy that generally allows Cubans who reach US territory to stay.
The original Cuban Adjustment Act of 1966 allowed Cubans to become permanent residents if they had been present in the United States for at least 2 years.
Those problems, officials contend, stem from the economic havoc wreaked by the embargo and are exacerbated by policies that allow Cubans who reach American soil to apply for asylum.
It was unclear how many people could be affected if the Administration allowed Cubans with families here or those convicted of only minor crimes to remain in the United States.