I have always been under the impression that our water in gravity fed from the Catskills up to the sixth floor as well...
95% of the water that is delivered to NYC from the Catskill/Delaware Watershed and the Croton Watershed is gravity-fed, so that's a good thing. The Croton water system was built back in the 1800's! Mechanical breakdowns, power outages within NYC, and high fuel costs don't impact that part of the system as much as in other cities that require a lot of pumping.
I'm not sure if water mains pressure alone is enough to refill rooftop water tanks, though, without the assistance of pumps. So, during a longer power outage, if you lived on a higher floor that generally relies on rooftop water tanks to maintain adequate pressure within the building, you may eventually "run out of" water on the higher floors because the rooftop tank finally ran dry and the water main pressure alone is not enough to push water to your floor.
Then again, during Queen's extended power outage last summer, I don't recall reading any news stories about people in taller apartment buildings have their taps run dry on higher floors, so I could be wrong. Hmm, or maybe those buildings all had non-electric backup pumps?