A CPR CAPT RPTED A READBACK HEARBACK PROB WITH ZKC WHICH RESULTED IN A TCASII TA.

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WHILE DSNDING THRU FL280 AT 200 FPM; (AUTOPLT VERT SPD ENGAGED); ATC CALLED; 'VERIFY ASSIGNED ALT IS FL290.' I AND THE 'PLT MONITORING' LOOKED AT THE ALT PRESELECTOR; WHICH READ FL240. THE 'PLT MONITORING' REPLIED; 'NO; WE WERE GIVEN FL240; AND THAT IS WHAT I READ BACK TO YOU.' I HAD BEGUN TO INSTINCTIVELY ARREST OUR DSCNT RATE; WHICH REACHED A STABILIZED ALT OF ABOUT FL274. I THEN NOTICED A 'TARGET' ON OUR TCASII DISPLAY; ABOUT 4 MI AWAY; 600 FT ABOVE OUR ALT; AND QUICKLY OBTAINED A VISUAL ON THE ACFT; AS DID THE 'PLT MONITORING.' PERHAPS SEPARATION (IFR) STANDARDS WERE COMPROMISED; BUT THERE WAS NEVER ANY DANGER OF A COLLISION. THEN; THE KANSAS CITY CTLR TOLD US TO 'MAINTAIN FL240.' WE ACKNOWLEDGED; AND CONTINUED OUR DSCNT. THEN; ZKC SAID; 'RECYCLE YOUR XPONDER; I'VE LOST YOUR MODE C.' (MIGHT THIS HAVE BEEN A 'CYA' ACTION?) WE 'RECYCLED;' AND THE REMAINDER OF THE FLT WAS UNEVENTFUL. WHEN WE RECEIVE AN ALT ASSIGNMENT; THE 'PLT MONITORING' SETS THE ALT PRE SELECTOR; READS BACK THE CLRNC TO ATC; CALLS OUT THE NEW ALT TO THE PF; AND THE 'PF' VERBALLY ACKNOWLEDGES THE ASSIGNED ALT. (BOTH PLTS WERE LISTENING TO THE ATC FREQ; NEITHER WERE DISTR BY OTHER DUTIES OR PAX). THE CTLR ISSUED US THE DSCNT TO FL240 IN ERROR. (WE HAD PREVIOUSLY BEEN CLRED TO FL290.) WE; AS A CREW OF 2; ACKNOWLEDGED A 'DSND TO FL240' CLRNC; THAT WAS GIVEN TO ANOTHER ACFT; THINKING AND HEARING IT WAS FOR US. WE READ IT BACK; AND SINCE WE WERE NOT CORRECTED OR CHALLENGED; WE COMPLIED WITH THE DSCNT. QUITE POSSIBLY; THE CTLR WAS WORKING MULTIPLE ACFT; AND DID NOT HEAR OUR READBACK. MAYBE THE CORRECT ACFT ACKNOWLEDGED THE DSCNT CLRNC ALSO; AND THE CTLR DID NOT HEAR US. MAYBE; THE CTLR WAS WORKING MORE THAN ONE FREQ; AND DID NOT HEAR OUR READBACK. (ANOTHER PROB OF MULTIPLE FREQS.) I HAVE BEEN IN CONTACT WITH THE ZKC FSDO!