IF you want to see EVERYTHING great there is too see youd have to live here, and I live here and I still havent seen everything there is to see... LOL But, to get to see most of the things that you NEED to see I would say ATLEAST one week.
(one week and a couple of hotels)
To see most of the island and spend a few hours on most of the beaches, 3 days should do it,
Sure, you can see it all in three days, from your car window... driving. You can also in three days see one or two major things but not all, especially if you count flight time in those three days.
Places to visit: the historical places in old San Juan (1 day at least at the forts, walking downtown which is very pinturesque, etc more if you are interested), visit the el Yunque Rainforest (1 day with commute), see the Camuy Caverns (1 day with commute), visit Ponce (Hacienda Buena Vista is a good historical place), the Arecibo Observatory (ever watch the movie Contact?), see the bio-fluorescent bay in Vieques (1 day with commute), maybe visit the Bacardi distillery if you are interested (close to old San Juan). How many days have I listed yet, and I haven't specifically counted for time for relaxing in the beach, which you may do in Vieques, by the way.
So I think the bottom line is that it's up to what you are interested to do the time you might want to spend in the island.
Just got back. We spent 3 nights which was OK, but we would have liked to have stayed longer. We ran out of time to do some of hte stuff we wanted to do - ride horses, kayak out to some of the small islands for snorkeling, etc. You could easily spend a week there....