TCASII RA SENDS ACR Y A CLB MESSAGE FOR A CLB TOWARDS OCCUPIED ALT. ACR X GETS TCASII RA FOR DSND MESSAGE TOWARDS OCCUPIED ALT. FLC DEV FROM ATC CLRNC.

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RECEIVING VECTORS FROM ZFW FOR GENERAL SEQUENCING INTO DFW OFF OF ACTON 3 ARR. DSCNT ISSUED BY CTR TO FL250 ON EASTERLY HDG. DURING DSCNT THROUGH FL252 GOT A 'TA' ON TCASII WITH TFC AT 12 O'CLOCK/7 MI AND APPROX MINUS 1400 FT IN A CLB. 'TA' FOLLOWED BY 'RA' CALLING FOR A DSCNT. COPLT (PF) CALLED OUT TFC IN SIGHT TO CREW; AND LEVELED OFF AT ASSIGNED ALT (FL250). TCASII ISSUED A SECOND RA -- 'INCREASE DSCNT; INCREASE DSCNT.' PF IGNORED ALL TCASII ADVISORIES AND INSTRUCTIONS. OTHER ACFT WAS OBSERVED IN A CLBING R TURN AND EVENTUALLY PASSED BY 9-10 O'CLOCK; 3- 4 MI AND MINUS 600 FT (FROM TCASII) (FL244). CTR LATER ASKED FOR OTHER ACFT TO CALL ZFW FOR POSSIBLE VIOLATION. AT NO TIME DID CTR CALL OUT EITHER ACFT AS TFC. NOTE: OTHER ACFT WAS CLBING TO LEVEL OFF AT FL240. ATC NEVER CALLED OUT TFC TO EITHER ACFT WITH OPPOSITE DIRECTION; HEAD-ON VECTORS WITH 1 ACFT DSNDING TO FL250 AND OTHER ACFT CLBING TO FL240. 'TA' WAS FIRST INDICATION OF POSSIBLE CONFLICT. CONFUSION IN COCKPIT AS TO RESPONSIBILITIES FOR FOLLOWING 'RA;' WITH OTHER ACFT IN SIGHT. APPARENTLY; OTHER ACFT DID NOT HAVE US IN SIGHT AND WAS FOLLOWING HIS TCASII RA TO CLB. TCASII APPARENTLY ASSUMED A 'XING CLB' SIT AND ATTEMPTED TO DIRECT ACFT THROUGH EACH OTHER'S ALT FOR RESOLUTION; WHEN ACFT WOULD NOT HAVE CROSSED NORMALLY. TCASII DIRECTED 2 INSTRUCTIONS WHICH WOULD HAVE CAUSED 2 DEVS FROM ASSIGNED ALTS AND; IN FACT; DID CAUSE ONE; AND ALSO CREATED A CONFLICT. CTR COULD HAVE AVOIDED CONFUSION BY IDENTIFYING ACFT TO EACH OTHER WITH ALT ASSIGNMENTS. SUPPLEMENTAL INFO FROM ACN 275507: RADAR VECTORED BY ZFW FOR A 'METERING DELAY' INTO DFW. SINCE WE WERE NOT IN VISUAL CONTACT WITH OTHER TFC; TCASII DIRECTIONS WERE FOLLOWED AND ACFT ALT REACHED FL244 BEFORE THE CAPT LOCATED OFFENDING TFC ABOVE AND TO OUR L. ATC HAD ERRONEOUSLY STATED THAT 'HE DIDN'T' KNOW WHY WE GOT THE TCASII ALERT SINCE BOTH ACFT HAD BEEN LEVEL FOR SOME TIME.' SUPPLEMENTAL INFO FROM ACN 275690: WE HAD BEEN LEVEL AT 24000 FT FOR SOME TIME AND TAKEN SEVERAL HDG CHANGES. ATC CAME RIGHT BACK AND REQUESTED THAT WE STOP CLBING; THAT WE HAD TFC AT 25000 FT. FLT STOPPED CLB AT 24500 AND RETURNED BACK TO 24000 FT. ATC THEN SAID ACFT HAD BEEN LEVEL 25000 FT AND THAT THERE SHOULD NOT HAVE BEEN TCASII RA. I SPOKE WITH THE CAPT OF CONFLICTING TFC LATER ON THE GND. HE INFORMED ME THAT THEY HAD DSNDED TO 25000 FT AT A PRETTY GOOD RATE OF DSCNT. OUR DELAY WAS OVER 15 MINS IN VARIOUS VECTORS; WE SHOULD HAVE BEEN PUT INTO HOLDING WHERE WE ALL HAVE A GOOD PICTURE WHERE ACFT ARE! MAYBE ATC SHOULD HAVE HELD US ON THE GND UNTIL HAVING A SLOT WITHOUT EXCESSIVE DELAYS INTO DFW. THERE IS TOO MUCH TFC IN THE DFW AREA TO BE RADAR VECTORING THAT MANY ACFT JUST TO DELAY FOR (GATE/FIX) TIME IN THE AIR.