B737-400 FLT CREW; UNAWARE OF NOTAM CONCERNING PARTIAL RWY AVAILABLE; DEPARTED FROM INTERSECTION IN OVERWEIGHT CONDITION.

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WE WERE #2 FOR DEPARTURE ON RWY 27R AT FLL. WE JUST RECEIVED OUR WT AND BAL. WE PROCEEDED TO APCH END OF RWY 27R. AS WE APPROACHED; THE CONTROLLER ASKED US; 'DID YOU GET BROWARD COUNTY PERMISSION TO OPERATE FROM FULL LENGTH?' WELL; THIS COMMENT TOOK US COMPLETELY BY SURPRISE AND CLOUDED MY SITUATIONAL AWARENESS. HE THEN TURNED US OVER TO GROUND CONTROLLER AND SHE STARTED TO GIVE US CONTACT INFORMATION FOR THE BROWARD COUNTY. WELL; AT THIS POINT THE FO AND I ARE FAR INTO THE 'YELLOW.' WE HAVE NEVER EXPERIENCED RECEIVING WT AND BAL FOR A PORTION OF THE RUNWAY THAT WAS RESTRICTED. APPARENTLY; DEPARTURES ARE ONLY PERMITTED FROM TXWY B5. WE FELT THAT PERHAPS; THE WT AND BAL WAS; IN FACT; BASED ON DEPARTURE FROM TXWY B5. WE FELT PRESSURED AND DECIDED THAT DEPARTURE FROM TXWY B5 WAS ACCEPTABLE (7700 FT REMAINING). WE DEPARTED WITHOUT INCIDENT. HOWEVER; ONCE AIRBORNE; WE TOOK A CLOSER LOOK AT THE WT AND BAL AND DETERMINED WE WERE OVERWT FOR A TXWY B5 INTERSECTION DEPARTURE. CALLED DISPATCH ON THE PHONE UPON OUR ARRIVAL IN ZZZ. I BRIEFED THE DISPATCHER ON THE EVENT AND ADMITTED WE TOOK OFF FROM INTERSECTION TXWY B5 IN AN OVERWEIGHT CONDITION. DEPARTURE SHOULD HAVE BEEN FROM FULL LENGTH. HE ADVISED ME THE NEXT FLIGHT FROM FLL EXPERIENCED THE SAME ISSUE BUT TOOK A DELAY TO GET PERMISSION; FROM BROWARD COUNTY; FOR A FULL LENGTH DEPARTURE. HE ALSO ADVISED ME A NOTAM WAS ISSUED ON THE PREVIOUS DAY THAT ONLY THE TOWER NEEDED TO BE ADVISED FOR A FULL LENGTH DEPARTURE. THIS NOTAM WAS NOT PRESENT IN OUR PAPERWORK. CRM; ERROR MANAGEMENT. IF WE HAD TAKEN MORE TIME; AND IGNORED OUR EXTERNAL PRESSURES; WE WOULD HAVE DEPARTED FROM FULL LENGTH. THIS WAS A 'TEXT BOOK' ERROR MANAGEMENT EVENT. I'M STILL KICKING MYSELF FOR RUSHING AND SUCCUMBING TO EXTERNAL PRESSURES (ATC; MULTIPLE AIRCRAFT WAITING FOR TAKEOFF). IN OUR DEFENSE; WE HAD MULTIPLE EVENTS; UNFAMILIAR TO US; TAKE PLACE IN A VERY SHORT PERIOD OF TIME. 1) WT AND BAL DATA FOR A RESTRICTED RUNWAY. 2) AN ATC TRANSMISSION THAT I FEEL WAS MORE A 'STATEMENT' (HEY KNUCKLEHEAD; WHERE ARE YOU GOING) THAN THAT OF A CLEARANCE.