A BOEING 737 FLC ACCEPTED A CLRNC INTENDED FOR ANOTHER ACFT AND CLBED ABOVE ASSIGNED ALT NEAR DBQ.

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ZAU INSTRUCTED ACR X TO SWITCH TO 132.22 FROM FREQ OF 135.7. RADIO XMISSION APPEARED WEAK. CAPT ACKNOWLEDGED; CTLR AGAIN XMITTED THE INSTRUCTIONS AND CAPT ACKNOWLEDGED AND SWITCHED TO 132.22. CAPT CHKED IN ON THIS NEW FREQ. A CTLR ACKNOWLEDGED ACR X AT FL330. THEN ACR X WAS INSTRUCTED TO CLB TO FL350. THIS WAS READ BACK BY THE CAPT AND ACKNOWLEDGED BY THE FO WHO WAS FLYING. THE FO SET FL350 IN ALT MCP. THE CAPT CHKED THE FMC CRZ PAGE TO CHK ALT CAPABILITY FOR WT AND TEMP. THE FMC INDICATED PROPER PERFORMANCE AND PROTECTION. FO USING LEVEL CHANGE CLBED TO FL350; AFTER BOTH CAPT AND FO SEARCHED THE TCASII RETURNS AND VISUAL SCANS OF AREA AHEAD OF ACR X. BOTH WONDERED ABOUT CLBING TO FL350 BUT HAD BEEN INSTRUCTED; ACKNOWLEDGED; CLRED AND FOUND NO IMMEDIATE CONFLICT; PROCEEDED AS INSTRUCTED. WAS NOT AWARE OR INFORMED OF ANY CONFLICTING CALL SIGNS OR TFC. AFTER APPROX 3-5 MINS AT FL350; WHAT APPEARED OR SOUNDED LIKE A DIFFERENT CTLR; ASKED ACR X 'WHEN DID YOU CHK IN.' THE CAPT REPLIED THAT 'ACR X HAD CHKED IN; AND HAD BEEN INSTRUCTED TO CLB TO FL350.' THE CTLR WAS SILENT. THE CAPT REPEATED THAT ACR X WAS AT FL350 AND WANTED ORD CTLR TO ACKNOWLEDGE. CAPT AND FO CONTINUED TO SCAN AIRSPACE AND TCASII RETURNS FOR ANY CONFLICTS. NONE OBSERVED. CTLR THEN SAID THEY WOULD GET US BACK DOWN TO FL330. ACR X WAS CLRED TO FL330 AND THEN HANDED OFF TO ANOTHER CTLR AT ORD. BEFORE HDOF CTLR SAID THAT THERE WAS POSSIBLE ACFT IDENT READBACK AND THE CLRNC COULD HAVE BEEN FOR AN ACR Y. HOWEVER PRIOR TO DSNDING US TO FL330; CTLR CLRED ACR Y TO FL350 AS IT APPEARED WE HAD PASSED OVER THAT FLT AND HE WAS BEHIND US.