BA46 AT FL290 DSNDS BELOW ASSIGNED ALT IN ZOB AIRSPACE.

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DURING CRUISE AT FL290; WE WERE GIVEN INSTRUCTIONS BY ZOB TO TURN R 20 DEGS. AS THE PNF; I HEARD TO TURN R 20 DEGS AND DSND TO FL280. I READ BACK THE CLRNC; 'TURN R 20 DEGS AND DSND TO FL280.' THE CAPT (PF) TURNED R 20 DEGS. WHEN I NOTICED THAT HE WAS NOT DSNDING; I TOLD HIM THAT THE CTLR WANTED US TO DSND TO FL280. DURING OUR DSCNT; I CALLED ZOB BACK AND ASKED IF HE WANTED US TO DSND TO FL280. THE FIRST TIME HE DID NOT REPLY. AFTER THE SECOND INQUIRY; THE CTLR SAID; 'NEGATIVE ACFT X; THERE IS AN ACFT UNDER YOU CLBING TO FL280.' DURING THE SECOND INQUIRY TO ATC; THE CAPT AND I NOTICED THE OTHER ACFT ON THE TCASII. THE TCASII WENT OFF WITH 'MONITOR VERT SPD;' FOLLOWED A FEW SECONDS LATER BY 'CLB; CLB NOW.' BY THIS TIME WE WERE BTWN FL286 AND FL285. THE CAPT IMMEDIATELY CLBED BACK TO FL290. THE CTLR TOLD THE OTHER ACFT TO STOP THEIR CLB; AND THEN CONTINUED THE CLB WHEN WE WERE BACK TO FL280. FLT TO DTW WAS THE LAST LEG OF THE 4TH DAY OF A CONTINUOUS DUTY OVERNIGHT. I ADMIT THAT I WAS TIRED FROM ONLY RECEIVING 4 HRS OF SLEEP AT THE HOTEL THAT MORNING. FATIGUE PLAYED A MAJOR ROLE IN THIS SIT.