ALT ASSIGNMENT CONFUSION IN ZID CLASS A AIRSPACE.

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EVENTS BEST AS I CAN RECALL. DEPARTED CLE; FO FLYING. CLRED BY DEP TO 17000 FT. CONTACTED ZOB; CLRED TO FL230; THEN FL280. CTR ASKED FOR CLB RATE OF 1500 FPM. WAS UNABLE; COULD DO 1000 FT. CTR ASKED US TO HURRY THROUGH FL220; THEN FL250. CLRED DIRECT PXV OUT OF FL240 FOR FL250. AT SOME POINT WE WERE HANDED OFF TO NEXT CTLR; ALONG WITH SQUAWK CHANGE AND CLRNC TO FL310. ACFT WAS IN SLOW CLB DUE TO HIGH GROSS WT AND ANTI-ICE. PNF WENT OFF FREQ TO MAKE PA AND RETURNED TO HEAR PF TALKING TO ZID. ZID ASKED IF WE WERE LEVEL FL280; TO WHICH HE REPLIED OUT OF FL280 FOR FL310. NO RESPONSE. SEVERAL OTHER ACFT CHKED IN. FINALLY WE ASKED TO CONFIRM ALT. CTLR STATED NO ONE DOWN HERE CAN REMEMBER CLRING YOU TO FL310. CALL REGARDING POSSIBLE DEV. WAS PRETTY SURE ZOB CLRED US TO FL310; BUT NOW WONDER IF WE GOT ANOTHER'S CLRNC. SIMILAR COMPANY CALL SIGNS ON FREQ THAT DAY. IN ADDITION; WAS LAST LEG OF A LONG DAY; AS WELL AS A BUSY FREQ WITH ORD TFC AND WX IN THE AREA. FOLLOWED ALT SETTING PROCS RIGOROUSLY; BUT NO GOOD IF I SET IT TAKING SOMEONE ELSE'S CLRNC OR CTR HDOF ISN'T COMPLETED. NOT A GREAT WAY TO END A TRIP.