A B737-300 FLT CREW IN THE VICINITY OF VHP VORTAC WAS CLRED TO DSND TO FL240. LEAVING FL342 THEY RECEIVED A TCAS TA FOLLOWED BY AN RA AND CLBED BACK TO FL340 TO AVOID CONFLICT WITH AN ACR CLBING TO FL330.

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AT FL350 APPROX 120 NM W OF THE BRICKYARD VOR WE RECEIVED AND ACKNOWLEDGED A DSCNT FROM ZKC TO WHAT WE BOTH BELIEVED WAS FL240. THIS WOULD ALSO CORRESPOND TO INTERCEPTING THE NORMAL DSCNT PROFILE INTO IND. FL240 WAS SET INTO THE FMC ALT WINDOW. AT APPROX FL342 I NOTICED A TARGET ON THE TCAS BELOW US CLBING ALMOST SIMULTANEOUS WITH A TCAS TA. I POINTED IT OUT TO THE CAPT (PF) AND HE BEGAN A LEVEL OFF THAT WAS IMMEDIATELY FOLLOWED BY A TCAS RA TO CLB AT APPROX 2000 FPM. THE LOWEST ALT THAT I SAW WAS FL338. I IMMEDIATELY ASKED CTR IF WE HAD BEEN CLRED TO FL240 OR FL340. THE CTLR RESPONDED '340.' AT THE SAME TIME ANOTHER CARRIER ACFT HAD BEEN CLBING TO FL330 AND RESPONDED TO A TCAS RA TO DSND. I STATED TO THE CTLR;' WE THOUGHT WE HAD RECEIVED CLRNC TO FL240 AND HAD ACKNOWLEDGED THAT PRIOR TO BEGINNING OUR DSCNT.' I DO NOT REMEMBER THE CTLR'S EXACT ANSWER BUT IT AMOUNTED TO A 'ROGER.' I ASKED IF THERE WAS A PROB AND THE CTLR ANSWERED 'NO.' THE CTLR ASKED THE OTHER CARRIER ACFT ABOUT RESUMING HIS CLB AND THE PLTS REITERATED THE TO ZKC CTLR THAT THEY HAD RESPONDED TO AN RA AND WERE NOW CLBING UP TO FL330. THE CTLR NEVER ASKED US ABOUT THE ALT MISUNDERSTANDING. THE CTLR EVENTUALLY CLRED US LOWER AND WITH A FREQ CHANGE SAID; 'THANKS FOR THE HELP.' BOTH THE CAPT AND MYSELF ARE NOT SURE WHAT HAPPENED. WE WERE CERTAIN OF OUR CLRNC TO FL240 BUT WE COULD HAVE BEEN WRONG. I GUESS WE WERE LOOKING FOR SOME TYPE OF ADMONISHMENT FROM THE CTR CTLR BUT THERE WAS NONE. IT LED US TO BELIEVE THAT THIS MAY HAVE BEEN AN ATC PROB. REGARDLESS; GOOD SITUATIONAL AWARENESS AS TO THE TCAS PRESENTATION AND BEGINNING A LEVELOFF AND OUR SUBSEQUENT RESPONSE TO THE TA/RA HAD US LEVEL AT FL340. THE TCAS TA AND THE SUBSEQUENT RA WAS RIGHT ON AND MINIMIZED A POTENTIAL SERIOUS CONFLICT. WE DO NOT BELIEVE THAT WE GOT CLOSER THAN A FEW THOUSAND FT TO THE OTHER CARRIER'S ACFT. WE WERE BOTH ALERT; CERTAIN OF OUR CLRNC; AND BOTH A LITTLE PUZZLED AS TO WHAT HAD ACTUALLY HAPPENED. NOT SURE WHAT CAN BE DONE ABOUT THESE COM MISUNDERSTANDINGS. WE ACKNOWLEDGED THE CLRNC WITH CTR; THE PNF (FO) PUT IT IN THE FMS WINDOW AND THE PF (CAPT) ACKNOWLEDGED THE ALT CHANGE; ALL ACCORDING TO PROC. NEEDLESS TO SAY WE DOUBLECHKED EVERY ALT CLRNC THE REST OF THE TRIP.