The weather can be as dangerous as others have said, so be prepared. On the other hand, in September the weather is more stable and you can get long periods of gorgeous weather. Sept. and Oct. are the best times for New England.
The end of one September, I hiked Mt. Adams, Madison and Jefferson staying at an RMC cabin. The weather was beautiful, sunny and 60F on top, but hikers who came over Mt. Washington told us that over night it got to 10F and 90 mph winds. We could hear the wind howling above us, but by the time we got above treeline both days it was really nice.