ALTDEV ALT EXCURSION DURING A TCASII RA.

Narrative

MY FO AND I WERE FLYING FLT FROM EL PASO TO DALLAS FT WORTH. THE FO WAS FLYING THE LEG AND WE WERE ESTABLISHED ON J-50; EBOUND AT FL250. AT APPROX PM13Z; 20-40 NM W OF WINK VOR; A YELLOW TA WAS ANNUNCIATED BY TCASII ON THE 20-MI RANGE. A QUICK LOOK AT THE MONITOR REVEALED A TARGET 2000 FT BELOW ON J-50 HEADED WBOUND. WITHIN MOMENTS; A SECOND YELLOW TA WAS ANNUNCIATED; THIS TIME; THE TARGET BEING AT 1000 FT BELOW. BOTH MY FO AND I WATCHED IT CLOSELY; ONLY TO OBSERVE A RED RA CONTINUING TO CLB UP AT US AND INDICATED 700 FT BELOW. ALSO; THE IVSI INDICATED A COMMAND TO CLB AT A RATE OF CLB OF AT LEAST 3500 FPM WHILE AN AUDIO 'CLB; CLB' CAME OUT OF THE TCASII SYS. I INSTANTLY COMMANDED MY FO TO CLB AT MAX RATE OF CLB TO AVOID THE TARGET. I INSTANTLY CALLED ATC TO ADVISE THEM OF THE DANGER. ATC RESPONSE WAS THAT THERE WAS NO TARGET THERE AT OUR FL; BUT THAT A MIL MTR WAS LEVELED BELOW US AT FL240. WE REACHED AN ALT OF APPROX 26000 FT WHEN RECOVERY WAS INITIATED AND OUR ACFT WAS RETURNED TO FL250. AT THIS POINT THE TARGET WAS VISIBLY BEHIND US ON TCASII AND IN THE CLR. ATC THEN INQUIRED A FEW MOMENTS LATER AS TO OUR RATE OF CLB. I REPLIED '3500- 4000 FPM.' ATC THEN INDICATED AN ALTDEV HAD TO BE RPTED. I REAFFIRMED TO ATC THAT WE INDEED HAD A DANGEROUS SITUATION; AND THAT TCASII HAD BOTH VISUALLY AND AURALLY COMMANDED A CLB. ATC SAID THE MTR WAS CLBING AT A HIGH RATE OF CLB; BUT NEVER LEFT FL240.