DURING WX DEVS AND HIGH TFC AND RADIO CONGESTION AN ACR ACCEPTS A CLB CLRNC MEANT FOR ANOTHER COMPANY ACFT WITH A SIMILAR CALL SIGN.

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LOCATION: DEVIATING N OF MIA 6 DEP; SKIPS TRANSITION; DEP UPON LINE-UP ON RWY 9L MIA. THE CAPT AND I NOTICED WX TO THE ESE. AFTER CONTACTING MIA DEP ON 125.5; I (AS PNF) PUT IN A REQUEST FOR WX DEV TO THE NNE. MIA DEP GAVE US OUR REQUESTED DEV AND EVENTUALLY HANDED US OFF TO ZMA. WE OBSERVED (BOTH VISUALLY AND ON RADAR) AN EXTENSIVE LINE OF TSTMS EXTENDING IN AN EASTERLY DIRECTION. RADIO COM WAS DIFFICULT DUE TO THE VOLUME OF TFC DEVIATING FOR WX. ATTEMPTS WERE MADE WITH ZMA TO REQUEST AN IMMEDIATE CLB. WE WERE STAIR-STEPPED TO FL200 AND DENIED FURTHER REQUESTS TO CLB. DUE TO THIS RESTR WE REQUESTED A HDG OF 070 DEGS TO AVOID WX. WITH NO RESPONSE FROM ZMA; WE AGAIN MADE THIS REQUEST. DURING A PERIOD OF HVY COMS WITH OTHER TFC; I THOUGHT I RECEIVED CLRNC TO TURN TO A HDG OF 070 DEGS AND CLB TO FL230. I READ BACK THIS CLRNC IN ITS ENTIRETY AND CONVEYED IT TO THE CAPT. WE BEGAN A TURN AND A SHALLOW CLB AS THE CAPT NOTICED PROX TFC ON TCASII. SHE ASKED FOR CONFIRMATION WITH CTR AS WE LEVELED OFF AT FL204. AT THAT MOMENT; ZMA RESPONDED TO MY READBACK AND DIRECTED A RETURN TO FL200. HE STATED THE CLRNC WAS FOR ANOTHER FLT. WE RETURNED TO FL200 AND CONTINUED TO REQUEST WX AVOIDANCE HDGS AT THAT ALT. ZMA WAS UNABLE TO RESPOND TO OUR REQUESTS AND EVENTUALLY ASKED US TO CONTACT THE NEXT CTLR WITH REQUEST. LESSONS LEARNED: BE MORE DIRECTIVE AND ASSERTIVE WITH ATC WHEN PRIORITY IS NEEDED. ENSURE CONFIRMATION OF REQUEST WITH ATC IS ACKNOWLEDGED EARLY IN THE PROCESS. CONTINUE TO ALWAYS COMPLY WITH READBACK OF INSTRUCTIONS. CONTINUE TO USE TCASII FOR SITUATIONAL AWARENESS. SUPPLEMENTAL INFO FROM ACN 541997: WE INITIATED THE TURN TO 070 DEGS AND STARTED THE CLB. PF SAW A TARGET ON THE TCASII THAT WAS 100 FT ABOVE AND ABOUT 9-8 O'CLOCK POS AT ABOUT 5 MI. PF STOPPED THE CLB AT FL204 AND ASKED THE PNF TO VERIFY CLRNC. PRIOR TO PNF VERIFICATION CALL; ZMA CONTACTED US AND DIRECTED US TO RETURN TO FL200 AND STATED THAT THE CLRNC TO FL230 WAS FOR ANOTHER FLT. WHAT I LEARNED AS PF AND PIC ON THIS FLT WAS TO BE MORE AGGRESSIVE IN REQUESTING DEVS FOR WX AND TO MAKE SURE THAT ATC UNDERSTANDS THE URGENCY OF MY DEV REQUESTS. JUST FOR YOUR INFO; AT NO TIME DID OUR TCASII SHOW ANY TA.