ACR DSNDS WHEN CLRED; CTLR LATER STATES CLRNC FOR SECOND ACFT WITH SIMILAR A/N.

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WE WERE CRUISING AT FL350 WHEN WE UNDERSTOOD ATC TO CLR US TO FL310 AND SLOW TO 270 KTS. (THIS IS COMMON FOR FLOW INTO MSP.) WE ACKNOWLEDGED THE XMISSION AND BEGAN OUR DSCNT. CTLR DID NOT ACKNOWLEDGE NOR CORRECT OUR READBACK. AT FL320 ATC CALLED TO SAY HE DID NOT CLR US TO FL310; BUT TO CONTINUE DSCNT TO FL310. HE THEN STATED HE HAD ANOTHER COMPANY WITH A SIMILAR SOUNDING CALL SIGN ON THE FREQ. (OUR FLT WAS AJBG; AND AABG ALSO ON FREQ.) COPLT AND I BOTH SAID TO EACH OTHER THAT HE HAD CALLED US WITH DSCNT CLRNC. WE QUERIED THE CTLR IF THERE WAS A PROB OR TFC CONFLICT WITH OTHER ACFT IN OUR VICINITY; AND REPLY WAS NEGATIVE. THIS SAME CTLR WAS ALSO WORKING 2 OTHER PAIRS OF OUR COMPANY FLTS WITH SIMILAR CALL SIGNS (AJFG; AAFG AND ANOTHER SET; SOMETHING LIKE IFI). CTLR WAS REALLY OVERWORKED WITH 6 COMPANY FLTS WITH SOUND-ALIKE SIGNS; AND AFTER OUR DSCNT HE BECAME VERY CLR AND CONCISE ABOUT THE CLRNCS. HE MAY HAVE CALLED US AND MEANT ANOTHER FLT; OR PERHAPS WE DID TAKE ANOTHER FLT'S CLRNC; BUT HE DID NOT ACKNOWLEDGE OUR RECEIPT OF THE CLRNC NOR CORRECT US AFTER OUR READBACK; THUS LEADING US TO BELIEVE IT WAS OUR CLRNC. MULTIPLE FLTS WITH SIMILAR SOUNDING SIGNS IN TODAY'S CONGESTED ATC ENVIRONMENT IS DANGEROUS; AND OUR COMPANY HAS A BAD PRACTICE OF DOING THIS. I BELIEVE THEY DO IT FOR MARKETING REASONS; BUT RUNNING BANKS OF FLTS INTO A HUB AT PEAK HRS WITH SIMILAR SOUNDING CALL SIGNS IS NOT A GOOD PRACTICE; AND SHOULD BE STOPPED; THUS HELPING TO AVOID SOMEONE FROM MISUNDERSTANDING AND TAKING SOME OTHER FLT'S CLRNC. THIS HAS THE POTENTIAL TO CREATE A VERY SERIOUS SIT. THIS CALL SIGN USAGE BY OUR COMPANY HAS RAISED THE IRE OF MANY PLTS; BUT OUR COMMENTS AND COMPLAINTS HAVE FALLEN ON DEAF EARS AT THE COMPANY.