B737 CREW STARTED CLB FROM 15000 FT TO FL230. THEY GET TCASII TA AND INCREASE CLB RATE TO AVOID CONFLICTING TFC.

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ACFT WAS LEVEL AT 15000 FT MSL JUST N OF THE RBS VOR ENRTE FROM ORD TO STL. CALL SIGN WAS 'AIRLINE XXX.' CAPT WAS PERFORMING THE DUTIES OF THE PF. THE FO WAS EXECUTING DUTIES OF THE PNF. MR X; AN FAA AVIATION SAFETY INSPECTOR WAS IN THE COCKPIT JUMP SEAT. AT APPROX XX40Z; AS I WAS REACHING INTO MY FLT BAG TO RETRIEVE THE STAR AND APCH PLATES I WOULD NEED FOR ARR INTO ST LOUIS LAMBERT FIELD; ZAU ISSUED A CLB TO FL230. FO ACKNOWLEDGED THE CLB AND RESET THE ALT ALERT WINDOW ON THE MODE CTL PANEL TO INDICATE 23000 FT. I POINTED TO THE NEW ALT SETTING; STATED 'CLBING TO FL230;' AND THEN INITIATED CLB USING VNAV CTL. PASSING APPROX 15500 FT WITH A CLB RATE OF ABOUT 1000 FPM; A TA WAS INITIATED BY THE TCASII SYS. THE INDICATION WAS AN ACFT AT ABOUT THE 10-11 O'CLOCK POS; LEVEL AT 16000 FT. I SPOTTED THE OTHER ACFT VISUALLY; IMMEDIATELY RECHKED MY ALT (15600 FT AND CLBING) AND CLB RATE. DUE TO THE JUXTAPOSITION OF BOTH ACFT AND THE CLB RATE VECTOR OF MY ACFT; I ELECTED TO INCREASE RATE OF CLB USING VERT SPD CTL TO 2000-2200 FPM. ESTIMATED HORIZ DISTANCE FROM THE OTHER ACFT AT THIS TIME WAS ABOUT 3 MI. NO RA WAS EVER ANNUNCIATED BY THE TCASII. AT 17500 FT I REDUCED THE CLB RATE TO 1000 FPM; RE-ENGAGED VNAV; AND CONTINUED CLB TO FL230. AFTER SWITCHING TO ANOTHER CTR FREQ; I WAS ASKED TO CALL ZAU. I ASKED MY FO IF; IN FACT; WE HAD BEEN CLRED TO FL230 AND IF HE HAD ACKNOWLEDGED THE CALL JUST TO CONFIRM IN MY MIND THAT WE HAD BEEN CLRED BY ZAU. HE STATED EMPHATICALLY THAT 'YES; HE HAD.' I ALSO ASKED THE FAA INSPECTOR IF HE HAD NOTICED ANYTHING OUT OF THE ORDINARY OR IF I HAD DONE ANYTHING ERRONEOUSLY. HE STATED; 'NO; I THINK YOU DID EVERYTHING YOU WERE SUPPOSED TO DO.' I HAVE NOT BEEN ABLE TO CONTACT ZAU QUALITY ASSURANCE/STANDARDIZATION AS YET TO DISCUSS THE MATTER WITH THEM. RADIO COMS BY ZAU WAS EXTREMELY BUSY WITH ALMOST CONSTANT XMISSIONS BTWN CTR AND OTHER ACFT IN THE AREA. ANOTHER ACFT -- DIFFERENT AIRLINE YYY -- WAS ALSO ON THE FREQ. I AM NOW THINKING THAT MY NEW FO; WHO WAS ONLY ON HIS THIRD FLT ASSIGNMENT FOR THE COMPANY; MIGHT HAVE ACKNOWLEDGED THE CLB INTENDED FOR DIFFERENT AIRLINE YYY INSTEAD OF US; AIRLINE XXX. I WAS TEMPORARILY DISTRACTED BY TRYING TO REMOVE THE APCH PLATES FROM MY FLT BAG AND MAY HAVE MISINTERPRETED/MISUNDERSTOOD THE CALL MYSELF DUE TO THE SIMILARITIES OF THE CALL SIGNS AND THE FRANTIC PACE AND CONSTANT CHATTER ON THE DEP OUT OF CHICAGO. THE FAA INSPECTOR DID NOT PICK UP ON ANY IRREGULARITY EITHER; BEFORE THE TA WAS INITIATED. THE ZAU CTLR ALSO DID NOT PICK UP ON THE FO'S CALL ACKNOWLEDGING THAT 'AIRLINE XXX' WAS LEAVING 15000 FT FOR FL230.