A CLBING B727 WITH CLRNC FROM ZOB TO FL310 HAS A POTENTIAL CONFLICT WITH XING TFC AT FL260. ZAU CTLR ATTEMPTS TO CLR CONFLICT BUT SAYS 'EXPEDITE THROUGH FL250.'

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I WAS THE CAPT OF FLT ON JUN/XA/00; DTW-STL. ZOB CLRED US TO FL310. SHORTLY THEREAFTER; ZOB HANDED US OFF TO ZAU. WE HEARD A GARBLED CTR XMISSION WITH EXPEDITE THROUGH FL250; BUT WERE UNSURE OF THE CALL SIGN. BEFORE WE COULD QUERY CTR; THE CTLR STATED 'EXPEDITE THROUGH FL250; OVER.' WE READ BACK THE CLRNC AND INCREASED OUR RATE OF CLB. PASSING FL250 I NOTICED A TCASII TARGET APCHING FROM THE S AT A HIGH RATE OF SPD. THE PROJECTED FLT PATH OF THE TARGET APPEARED TO CONFLICT WITH OURS. AS I KNEW IT WAS NOT POSSIBLE TO CHANGE OUR RAPID ASCENT INTO A DSCNT WITHOUT PASSING FL260 I INSTRUCTED THE FO TO AGAIN INCREASE OUR RATE OF CLB SO AS TO PASS THROUGH FL260 PRIOR TO OUR FLT PATH INTERSECTING THE FLT PATH OF THE OTHER ACFT. I ALSO INSTRUCTED HER THAT IF NECESSARY TO AVOID THE ONCOMING ACFT WE WOULD TURN S. WE CLRED THE OTHER ACFT BY 300 FT VERTLY AND 2 MI HORIZLY. CTR THEN BECAME BUSY ATTEMPTING TO RESOLVE THE PROB AND STATED THAT SOMEONE HAD 'BLOWN THROUGH' THEIR ALT. HE THEN ASKED US OUR ALT WHICH WAS APPROX FL270 CLBING TO FL310. HE STATED WE HAD NOT BEEN CLRED TO FL310. PRIOR TO LEAVING THE FREQ I OBTAINED THE APPROPRIATE PHONE NUMBER TO CONTACT WHEN WE ARRIVED STL. I CONTACTED ZAU VIA TELEPHONE FROM STL; IDENTED MYSELF; AND SPOKE WITH ATC SUPVR. HE ATTEMPTED TO LOCATE THE CTLR BUT WAS UNABLE TO. I EXPLAINED THE NATURE OF THE CALL. HE INFORMED ME THE ATC TAPES WERE BEING REVIEWED AS WE SPOKE BUT THE MAIN QUESTION WAS 'WHERE DID WE RECEIVE OUR CLRNC TO FL310? HE INSTRUCTED ME TO CALL HIM BACK LATER. I CALLED BACK SEVERAL HRS LATER AND WAS INFORMED THE TAPES HAD BEEN REVIEWED. THE END RESULT APPEARS TO BE LACK OF COM BTWN CTRS. PRIOR TO FLT DEPARTING CLEVELAND'S AIRSPACE; CHICAGO CALLED CLEVELAND VIA LANDLINE AND ASKED THEM TO RESTRICT US TO FL250. THIS WAS NOT ACCOMPLISHED. WHEN WE CONTACTED CHICAGO WE TOLD THEM WE WERE CLBING TO FL310. FOR WHATEVER REASON; THE CTLR FAILED TO COMPREHEND WE HAD A CLRNC ABOVE FL250. ACCORDING TO SUPVR; THE CHICAGO CTLR ASKED US TO EXPEDITE 'TO' FL250 NOT 'THROUGH' FL250. SUPVR ADVISED ME THE SIT WOULD BE HANDLED IN-HOUSE AND THE FAA WOULD NOT BE TAKING ACTION AGAINST ME OR COMPANY. IF OUR ACFT HAD NOT BEEN EQUIPPED WITH TCASII AND IMC CONDITIONS WERE PRESENT; THE RESULT COULD HAVE BEEN CATASTROPHIC. I BELIEVE THE CAUSE OF THIS INCIDENT WAS FAILURE OF COMPLETE COM BTWN CTRS AND THE CHICAGO CTLR NOT COMPREHENDING OUR FIRST XMISSION WHEN WE STATED WE WERE CLBING TO FL310.