A B737 FLT CREW THOUGHT IT WAS CLRED TO FL350; BUT LATER ATC INFORMS THEM THAT THEY TOOK A CLRNC FOR ANOTHER ACFT.

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ON CLBOUT FROM STL WE WERE GIVEN A CLB TO FL280. WE REQUESTED FL350 FOR A FINAL. WE WERE GIVEN CLB TO FL290 THEN A CLB TO FL350 AFTER WE HAD BEEN LEVEL AT FL290. ALL CLRNCS WERE READ BACK 'COMPANY # AND THE ALT CLRED TO.' AT FL305 A DIFFERENT CTLR CAME ON AND ASKED OUR ALT. WE RESPONDED 'COMPANY # AND GAVE OUR ALT AND THAT WE WERE CLBING TO FL350.' THE FIRST CTLR STARTED TO ISSUE US A DSCNT AND THE SECOND CTLR SAID 'NO' IN THE BACKGROUND. WE CONTINUED AT FL350 FOR ABOUT 1 HR WHEN CTR GAVE US A NUMBER TO CALL ABOUT AN ALTDEV. NOTHING ELSE HAD BEEN SAID TO US ABOUT THIS. ACCORDING TO ATC; WE TOOK A CLB CLRNC FOR ANOTHER ACFT. WE NEVER HEARD ANOTHER ACFT RESPOND TO THE CLRNC AND WE NEVER KNEW THIS WAS A SIMILAR CALL SIGN ON THE FREQ. THE CAPT AND I WERE NOT TIRED OR DISTRACTED IN ANY WAY; AND WE BOTH THOUGHT WE HAD BEEN CLRED TO FL350. IT NEVER OCCURRED TO US THAT WE MAY HAVE DONE ANYTHING WRONG UNTIL WE RECEIVED THE MESSAGE TO CALL ATC ABOUT A POSSIBLE ALTDEV. AS FAR AS WE KNEW; WE HAD COMPLIED WITH THE CLB INSTRUCTIONS.