AN ARR ATR72 FLC LEAVES THEIR ASSIGNED ALT OF 5000 FT WHEN RECEIVING A TCASII RA. WHILE ADVISING APCH CTLR OF SAME; CTLR ADVISES CREW TO RETURN TO 5000 FT BECAUSE OF XING TFC AT 6000 FT. MISS DISTANCE WAS 3000 FT HORIZ AND 550 FT VERT.

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WHILE ON APCH TO RWY 17C AT DFW; TCASII GAVE AN IMMEDIATE 'CLB NOW' RA WITH VSI INDICATION OF 1500 FPM CLB. WHILE RESPONDING TO THE RA; WE CLBED TOWARDS AN MD80 1000 FT ABOVE OUR ALT. NOTIFIED ATC WE WERE RESPONDING TO THE RA. WE WERE TOLD OF THE CONFLICT; RETURNED TO OUR ASSIGNED ALT OF 5000 FT. TOTAL DEV OF 500 FT. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE CAPT RPTR; PNF; COULD NOT TELL THE ANALYST WHAT MAKE OF TCASII THEY HAD INSTALLED ON THE ATR72. HE DOES; HOWEVER; BELIEVE IT WAS A FALSE WARNING; A GHOST. THE TFC WAS INDICATED AT THE SAME ALT AS THEIR ACFT AND THE TCASII HAD NOT ISSUED A TA PRIOR TO THE ISSUANCE OF THE RA. ATC DID NOT SEE ANY TFC BELOW OR AT THE SAME ALT. AT THAT TIME THOUGH; THERE WAS A LOT OF TSTM ACTIVITY IN THE AREA; WHICH WAS A DISTR TO BOTH THE FLCS AND ATC. THE ACFT WAS IMC AT THE TIME AND THAT WAS ONE REASON FOR THE ALTDEV ALT EXCURSION AND NON ADHERENCE TO ATC CLRNC. THE CAPT NOTED TFC 'EVERYWHERE' AND THEY WERE ON VECTORS AFTER A HECTIC WX WHIPPED FLT FROM LITTLE ROCK. THE CREW WAS TASK SATURATED. THE CAPT WAITED FOR THE THIRD RA BEFORE HE INSTRUCTED THE FO TO CLB WITH THE RA SINCE HE FELT THAT THE TCASII SOMETIMES SETTLES DOWN AFTER A FALSE 'BLURP' OR TWO. THE CAPT ADVISED ATC WHO JUST ABOUT 'CAME OUT OF HIS CHAIR' AND SAID; 'DON'T CLB; THERE'S XING TFC ABOVE YOU AT 6000 FT.' THE ATR72 THEN POPPED OUT OF THE CLOUD AT ABOUT 5500 FT AND SAW THE XING TFC. A WK LATER THE CAPT CALLED THE TRACON SUPVR AND DISCUSSED THE MATTER WITH HIM. THE SUPVR SAID THAT THE ACFT AT 6000 FT DID GET AN RA ON THE ATR72 AND HAD STARTED A CLB AS WELL. A REAL BUMP UP OP. SUPPLEMENTAL INFO FROM ACN 415626: WE NOTIFIED ATC OF THE RA AND DEV AND WAS INSTRUCTED TO DSND IMMEDIATELY FOR XING TFC.