2 ACR ACFT HAD LTSS. NEITHER ACFT FOLLOWED ITS TCASII COMMAND.

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TCASII TOLD US TO CLB WHEN WE WERE SUPPOSED TO LEVEL AT FL260. OUR INFO SAYS 'ACR MLG 2' GOT A DSND COMMAND. IT APPEARS ACR 2 WAS ABOVE US CLBING AND WE WERE CLBING SO FAST TO FL260 THAT THE TCASII DECIDED WE WOULD CLB AND ACR 2 SHOULD DSND. WE THINK ACR 2 STARTED TO DEVIATE ON A TCASII ALERT AND THEN STOPPED. HE DID ANNOUNCE HE WAS DEVIATING ON A TCASII ALERT. IT APPEARS THAT; IF WE HAD CLBED AS PER THE INSTRUCTIONS; WE WOULD HAVE CLBED RIGHT AT ACR 2. TCASII RPT FILED WE THINK BY ACR 2 AND BY ZME. SUPPLEMENTAL INFO FROM ACN 236477: WE WERE AT 26000 FT IN CRUISE AND RECEIVED A CLB CLRNC TO 28000 FT. AS WE WERE CLBING THROUGH APPROX 27300; WE GOT A TCASII 'TA' AND THEN ALMOST IMMEDIATELY WE GOT AN 'RA' WITH 'DON'T CLB' AND A 1500 FPM DSCNT COMMAND. THERE WAS AN ACR 1 MLG APPROX 3 MI ON OUR R AT 26000. HE HAD BEEN GIVEN A CLB TO 27000; AND HE TOO WAS RECEIVING AN RA THAT WAS COMMANDING A CLB. WE STOPPED OUR DSCNT AT 26700 BECAUSE I COULD SEE THE OTHER ACFT WAS BELOW US. IF WE BOTH HAD FOLLOWED THE TCASII COMMANDS; I FEEL WE WOULD HAVE BEEN IN A VERY DANGEROUS SIT. THE CTLR SAID HE WAS GLAD THAT WE STOPPED OUR DSCNT BECAUSE THE OTHER ACFT WAS BELOW US. LUCKILY; THE OTHER ACFT DIDN'T CLB TOO MUCH; OR I THINK WE WOULD HAVE BEEN REALLY CLOSE TO EACH OTHER.