AN MD88 FLC FAILED TO MAINTAIN VISUAL SEPARATION WITH A B767 AND HAD AN NMAC NEAR PMM.

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WE WERE ON THE RADAR VECTORS NE OF THE PULLMAN ARR PASSING THROUGH FL180 WHEN WE RECEIVED A TCASII TA FOLLOWED BY AN RA COMMANDING US TO REDUCE DSCNT RATE. WE COMPLIED WITH THE RA COMMAND AND LEVELED OFF AT OUR ASSIGNED ALT OF 16000 FT. ATC GAVE US A CLRNC TO DSND TO 10000 FT. A FEW SECONDS LATER WE GOT ANOTHER TA FOLLOWED IMMEDIATELY BY AN RA. THE CAPT INITIATED A CLB TO COMPLY WITH THE RA WHILE I INFORMED ATC THAT WE WERE UNABLE TO COMPLY WITH THE DSCNT CLRNC. COMPANY Y FLT HDG SE CROSSED OUR FLT PATH CLOSE IN FRONT OF US AT OUR ALT. A FEW SECONDS LATER WE RECEIVED THE TCASII 'CLR OF CONFLICT.' WE QUESTIONED ATC AND WERE TOLD THAT THE ACR FLT HAD US IN SIGHT AND HAD REQUESTED A VISUAL CLB AND ATC HAD GRANTED THE REQUEST; TELLING THE COMPANY Y FLT TO MAINTAIN VISUAL SEPARATION FROM US. THE COMPANY Y PLT RESPONDED TO ATC AND TOLD US THAT HE HAD MISJUDGED HIS RATE OF CLB AND HIS SEPARATION FROM OUR FLT; AND HE APOLOGIZED. WE CONTINUED OUR DSCNT AND APCH INTO ORD. THE APCH AND LNDG WERE UNEVENTFUL. THE COMPANY Y JET WAS CLBING INTO US AND PASSED FROM THE LOWER R SIDE OF OUR ACFT THROUGH THE UPPER L SIDE OF OUR ACFT; PASSING OUR ALT DIRECTLY IN FRONT OF US. SINCE WE WERE RESPONDING TO AN URGENT CLB RA; WHAT KIND OF TCASII INFO WAS THE OTHER CREW GETTING? WERE THEY ALSO GETTING A CLB RA THAT WAS PUTTING US ON A COLLISION COURSE? OR WERE THE PLTS OF THE COMPANY Y JET IGNORING THEIR TCASII INFO? IT IS MY UNDERSTANDING THAT; ACCORDING TO COMPANY Y FLT OPS SPECS; THEY DO NOT HAVE TO TAKE ACTION WHEN A TCASII RA IS RECEIVED; UNLIKE COMPANY X'S OPS SPECS; WHICH STATES THAT IT IS MANDATORY THAT WE FOLLOW THE RA UNLESS WE HAVE VISUAL CONFIRMATION THAT THE RA WOULD PUT THE ACFT IN DANGER. THE ENTIRE INCIDENT BEGAN IN CLASS A AIRSPACE. IS IT A SAFE PRACTICE TO ASSIGN AN ACFT VISUAL CLRNCS IN CLASS A AIRSPACE? ESPECIALLY IF THE OTHER ACFT DOES NOT HAVE YOU IN SIGHT? WHERE WAS THE CTLR IN ALL OF THIS? NOT ONCE DID THE CTLR ISSUE A TFC CONFLICT WARNING TO OUR FLT. NOT UNTIL AFTER THE INCIDENT; WHEN I QUERIED THE CTLR; DID ANY HINT OF CONCERN COME FROM ATC; EVEN THOUGH TFC WAS VERY LIGHT AT THE TIME IN OUR SECTOR; AND RADIO TFC WAS PROPORTIONALLY LIGHT. IN MY OPINION; THE INCIDENT WAS CAUSED BY THE COMPANY Y PLT'S INABILITY TO CORRECTLY JUDGE DISTANCE AND MOVEMENT OF OUR ACFT IN RELATION TO THEIR OWN. A CONTRIBUTING FACTOR WAS THE VISUAL CLB CLRNC GRANTED BY ATC; AND THE LACK OF IMPORTANCE ASSIGNED TO THE TCASII WARNINGS THE COMPANY Y JET MUST HAVE BEEN GIVEN. SUPPLEMENTAL INFO FROM ACN 453658: THE CTR ASKED US IF WE HAD VISUAL CONTACT WITH TFC AT 10 O'CLOCK POS. WE DID HAVE THE TFC IN SIGHT. WE ACKNOWLEDGED VISUAL CONTACT; AND ACCEPTED CLRNC TO CLB REF THAT COMPANY X TFC. WE THOUGHT WE WERE CLBING THROUGH COMPANY X'S ALT BEHIND HIS WING LINE -- PROBABLY SEPARATED BY LESS THAN RADAR REQUIREMENTS. I DETERMINED VISUALLY THAT OUR SEPARATION WAS DECREASING; AND THAT OBVIOUSLY OUR TRACKS WERE NOT PARALLEL. MY RESPONSE WAS TO ESTABLISH AN AGGRESSIVE CLB IN EXCESS OF 5000 FPM. WITH THAT RATE OF CLB IN PROGRESS; WE RECEIVED A TCASII TA/RA COMMANDING A DSCNT. AT THAT POINT; OUR ALT WAS JUST ABOUT LEVEL WITH THE CONFLICT TFC. I CHOSE TO IGNORE THE RA 'DSND' COMMAND BECAUSE VISUALLY I COULD DETERMINE THAT SUCH A MANEUVER; TRANSITIONING FROM A VERY RAPID CLB TO THE TCASII DEMANDED DSCNT; WOULD TAKE US ABOVE COMPANY X THEN BACK DOWN IN FRONT OF THEIR PATH. ADDITIONALLY; TO CHANGE FROM SUCH A HIGH RATE OF CLB TO THE TCASII DSCNT WOULD HAVE BEEN A DANGEROUS MANEUVER FOR MY ACFT. SO I CONTINUED THE CLB. THE COMPANY X CREW ANNOUNCED THAT THEY HAD AND WERE RESPONDING TO A TCASII RA; WHICH APPEARED TO BE A CLB; AND FURTHER REDUCED OUR SEPARATION.