B737-300 FLT CREW IS CLRED INTO POS FOR TKOF AT FLL. MID SENTENCE; LCL CTLR; HAVING ISSUED HDG AND ALT AFTER DEP; STOPS. FLT CREW ASSUMES CLRNC FOR TKOF AND DEPS. THEY ARE DELAYED BY MIA DEP DUE TO TFC VOLUME IN ZJX AIRSPACE; BUT EVENTUALLY ARE CLBED ON COURSE TO DEST.

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THE FLT WAS THE LAST OF A 3-DAY PAIRING; THE FIRST OF 5 LEGS. WE WERE NOTIFIED WHEN WE ARRIVED THAT ZJX WAS DOWN AND DISPATCH HAD FILED US AT 9000 FT AND ADDED ADDITIONAL FUEL FOR THE FLT. WE WERE ADVISED BY CLRNC DELIVERY THAT ZJX WAS BACK UP AND RECEIVED THE STANDARD COMPANY CLRNC. FLL TFC HAD SIGNIFICANT DELAYS. FLL GND AND TWR WERE CLRLY FAR BUSIER THAN NORMAL. IT TOOK US APPROX 30 MINS TO REACH THE DEP RWY (RWY 27R). WE WERE CLRED 'ON TO HOLD' EXPECTING MINIMUM TIME ON THE RWY; AS TFC WAS APPROX 4-5 MI ON FINAL. WITHIN LESS THAN 1 MIN; TWR MADE A NONSTANDARD XMISSION. TWR APOLOGIZED FOR THE DELAY ON THE GND; PROVIDED WINDS AND INITIAL DEP HDG; BUT NEVER COMPLETED THE XMISSION WITH 'CLRED FOR TKOF.' WITH THE START OF THE XMISSION; THE CAPT ADDED PWR AND BEGAN THE TKOF RUN TO AVOID CONFLICT WITH TFC ON FINAL. BY THE TIME I REALIZED THE WORDS; 'CLRED FOR TKOF' WERE NOT IN THE XMISSION; WE WERE ACCELERATING DOWN THE RWY. I READ BACK THE DEP HDG AND 'CLRED FOR TKOF' IN THE HOPE OF CLARIFICATION FROM TWR. TWR THEN XMITTED SOMETHING TO THE EFFECT OF 'YOU'RE ALREADY ROLLING.' ONCE AIRBORNE; WE WERE SWITCHED OVER TO DEP WITHOUT FURTHER COMMENT. MIA DEP INITIALLY DENIED CLRNC TO THE HIGH CTR ALTS. AFTER A SHORT DELAY; WE WERE CLRED INTO CTR AIRSPACE AND PROCEEDED TO OUR DEST. THIS IS WHAT I 'THINK' HAPPENED: AS FLL TWR WAS IN THE PROCESS OF CLRING US FOR TKOF; HE RECEIVED NOTIFICATION FROM DEP CTL TO HOLD US ON THE GND DUE TO CTR SATURATION. THIS CAUSED HIM TO STOP HIS TKOF XMISSION IN MID-SENTENCE. HE DID NOT TAKE ACTION TO STOP OUR TKOF BECAUSE WE WERE ALREADY ACCELERATING DOWN THE RWY. SUPPLEMENTAL INFO FROM ACN 605187: WE WERE THEN CLRED ONTO THE RWY TO HOLD. 'TFC ON A 5 MI FINAL; BE READY.' (I THINK I HEARD CLRED FOR TKOF.) THE FO READ BACK; 'HDG 270 DEGS; 3000 FT CLRED FOR TKOF' AND WE ROLLED. TWR THEN SAID; 'OH; I SEE YOU'RE ON THE ROLL.' I THOUGHT NOTHING OF THIS; AS WE CLBED TO 3000 FT. WHEN WE CONTACTED MIA DEP; THEY INFORMED US ZJX WAS AGAIN NOT ACCEPTING TFC; AND SHE WOULD HAVE TO KEEP US IN HER AIRSPACE. LONG STORY SHORT; THEY WORKED IT OUT AND WE WERE ON OUR WAY. THINKING BACK; TWR MAY HAVE RECEIVED WORD FROM ATC TO STOP DEPS AND MAY NOT HAVE GIVEN TKOF CLRNC. I REALLY DON'T KNOW. I'LL MAKE SURE THE NEXT TIME.