ZOB CTLR EXPERIENCED OPERROR AT FL210 AFTER ASSIGNING FL270 TO A CLBING ACFT WHEN THE RECEIVING CTLR WAS EXPECTING THE ACFT AT FL200.

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C525 DEP ASSIGNED CLB TO FL270. POINTOUT MADE TO ADJACENT SECTOR WHEN TFC WAS SEEN (FL210). I COORD THE C525 CLBING TO FL200. I THEN COORD TO ANOTHER SECTOR IN ANOTHER FACILITY THE C525 CLBING TO FL200. I INADVERTENTLY ASSIGNED FL270 TO THE C525 AND WHEN I SWITCHED THE AIRPLANE TO THE NEXT FREQ (ANOTHER FACILITY) THE C525 CONTINUED TO CLB THROUGH THE TFC (AN E145 AT FL210). NEITHER ACFT WAS ON MY FREQ TO TURN OR ASSIGN ALTS AT THE TIME THE LOSS OF SEPARATION OCCURRED. SUPPLEMENTAL INFO FROM ACN 644195: WE WERE ASKED TO CLB TO FL200 AND BE THROUGH FL190 IN 1 1/2 MINS. OUR CLB THROUGH FL190 WAS LESS THAN 1 MIN; THEN LEVEL FL200. THEN CLEVELAND SAID TO FL270; THEN AMENDED TO FL230; THEN AMENDED TO FL200; THEN CLE SAID 'CLB TO FL270 AND CONTACT NEW YORK 132.42?' WE CHKED IN WITH NEW YORK. WHILE CLBING TO FL270; NEW YORK SAID 'DSND TO FL220 RIGHT AWAY.' WE STARTED DOWN OUT OF FL21-220. THEN NEW YORK SAID 'TFC; TAKE WHATEVER YOUR TCASII SAYS TO DO.' I SAW THE TFC WAS BELOW; SO I STARTED A CLB AND TURNED R. NEW YORK THEN SAID 'CLB TO FL270. THERE WAS ANOTHER ACFT WITH A SIMILAR CALL SIGN ON FREQ. TOO MANY ALT CHANGES AND SIMILAR CALL SIGNS ON FREQ. BOTH MYSELF AND FO HEARD AND READ BACK OUR FULL CALL SIGN.