A CL65 RECEIVE A GPWS WARNING WHEN THEY HEARD AND READ BACK AN ALT ASSIGNMENT WHICH ENDED UP BEING 1000 FT LOWER THAN INTENDED.

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GOING INTO SWF; NEW YORK APCH GAVE US A HDG OF 170 DEGS AND 6000 FT. ONCE ON THE HDG OF 170 DEGS; ATC TOLD US TO DSND AND MAINTAIN 3000 FT. I READ BACK THE CLRNC 'DSND AND MAINTAIN 3000 FT.' THERE WAS NO CORRECTION OF MY READBACK; SO WE SET 3000 FT AND CONFIRMED AND BEGAN THE DSCNT. AT 4000 FT; WE CONFIRMED 1000 FT TO GO AS PER COMPANY POLICY. AT APPROX 3450 FT; WE RECEIVED A GPWS 'TERRAIN.' MY FO PERFORMED THE ESCAPE MANEUVER AND SIMULTANEOUSLY WE RECEIVED A XMISSION FROM ATC 'ACR X YOU WERE CLRED TO 4000 FT.' WE LEVELED AT 4000 FT AND LANDED WITH NO FURTHER INCIDENT. ONCE ON THE GND AT THE GATE MY FO ASKED THE GND CTLR IF APCH CTLR WAS IN TRAINING. THE REPLY WAS NO; THEY WERE JUST EXTREMELY BUSY. I KNOW THAT AREA IS VERY BUSY WITH A LOT OF ACFT AND WHEN THE WX MOVES IN; IT CAN BE A MESS. I BELIEVE ALL IN ALL; THE CTLRS DO A WONDERFUL JOB; I AM NOT AWARE OF THEIR PROCS. DO THEY HAVE SOMEONE HELPING OR WATCHING THEM ESPECIALLY WHEN IT GETS HECTIC? AS FOR MY CREW; WE WERE FRESH; FIRST LEG OF THE TRIP. WE HAD FLOWN 1 TRIP TOGETHER BEFORE THIS ONE. WE BOTH HEARD 3000 FT AND FOLLOWED PROCS FOR OUR COMPANY. IF WE DID ACTUALLY HEAR INCORRECTLY; WHICH ATC'S CALLS 'YOU WERE CLRED TO 4000 FT' AND THE GPWS 'TERRAIN' CALL WOULD MAKE US BELIEVE. POSSIBLY CLOSER MONITORING WOULD HELP.