FLT OVERSHOT ASSIGNED ALT IN DSCNT; DSNDED THROUGH AN OCCUPIED ALT AND GOT A TCASII RA.

Narrative

THE FLT PLAN CALLED FOR ROUTING FROM JOT TO DNV AND THE MILAN 1 ARR. THE ARR FROM OVER DNV TO SHB IS NOT DIRECT AND CONTAINS A SOUTHERLY AND THEN EASTERLY ROUTING. ZAU CLRED OUR FLT DIRECT FROM A POINT APPROX 30 MI SE OF JOT DIRECT TO SHB THUS AVOIDING MOST OF THE ARR. THE FLT WAS PROCEEDING AS CLRED WHEN THE CTR ISSUED A 20 DEG HDG TO THE S TO AVOID A MIL TRAINING AREA. SHORTLY AFTER THIS A CHANGE IN ALT WAS GIVEN AS WELL AS A CLRNC TO PROCEED ONCE AGAIN TO SHB. AS WE INITIATED A SLOW DSCNT; TO WHAT WE BELIEVED WAS THE CLRED ALT; THE CTR ISSUED A FREQ CHANGE. PRIOR TO INITIATING CONTACT ON THE NEW CTR FREQ; OUR TCASII ALERTED WITH A TA. AS OUR ATTN WAS CALLED TO THE 'ALERT' THE TCASII WENT INTO AN RA. AS WE ANALYZED AND BEGAN TO REACT TO THE RA; THE CTR CALLED ASKING OUR ALT AND IMMEDIATELY ISSUED A R TURN TO INCREASE SPACING WITH THE TCASII TFC. THE CTR ASKED OUR ALT AND IT WAS AT THIS POINT WE WERE INFORMED WE WENT THROUGH THE ALT ISSUED BY THE LAST CTR. THE CTR SAYS THAT WE WERE CLRED TO FL270; THE CREW BELIEVES WE WERE CLRED TO FL240. THE ALT OF FL240 WAS IN OUR ALT ALERT BOX. THE PROC IS FOR THE PNF TO SET THE ALT ISSUED IN THE BOX AND POINT TO IT TILL THE OTHER PLT CONFIRMS THE ALT. WE HAD DONE THIS PROC THE PREVIOUS 2 DAYS AS WE ALWAYS DO; SO WE BELIEVE THAT WE PROPERLY SET THE ALT AND FOLLOWED THE PROC.