CLRNC INTERP LEADS TO ALTDEV ALT EXCURSION BY AN A320 FLC. TSTM ACTIVITY CITED AS CAUSAL.

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THERE WAS A LOT OF TSTM ACTIVITY IN THE VICINITY OF THE ARR ARPT. THIS CREATED CONSIDERABLE RADIO FREQ CONGESTION. WE HAD NUMEROUS REQUESTS FOR HDG CHANGES TO FLY AROUND CELLS. WE WERE AT 8000 FT ON A MODIFIED DOWNWIND. ATC CLRED US TO 6000 FT; SHORTLY THEREAFTER HE WANTED US TO SLOW TO 170 KTS. WE ACKNOWLEDGED AND COMPLIED. I THEN REQUESTED A L TURN; 070 DEGS APPROX. APCHING 7000 FT ATC APPARENTLY INQUIRED OR REQUESTED IF WE COULD HOLD 7000 FT. I DID NOT HEAR THIS; ACKNOWLEDGE THIS; NOR RESET THE ALT ALERT ON THE FLT CTL UNIT. ACFT WAS ON AUTOPLT. APPROX 6500 FT ATC INQUIRED IF WE WERE GOING TO MAINTAIN 7000 FT. FO RESPONDED; WE DID NOT HEAR THE AMENDED ALT. ATC THEN TOLD US TO MAINTAIN OUR PRESENT ALT. WE REMAINED AT 6500 FT. I BELIEVE THE MISUNDERSTANDING OF OUR CLRED ALT WAS A RESULT OF THE BUSY CONDITIONS; IE; WX AND TFC; AS WELL AS MY REQUEST OF A 70 DEG HDG; ATC REQUEST FOR 170 KTS AND AT THE SAME TIME THE INQUIRY TO MAINTAIN 7000 FT. ALSO A BETTER CONFIRMATION BY ATC AS TO OUR UNDERSTANDING OF THE CLRNC AND WAITING FOR THE PROPER READBACK. OTHER FACTORS INVOLVED WERE MY CONCENTRATION AND FOCUS ON AVOIDING THE TSTMS IN OUR PATH. PRIOR TO GOING OVER TO THE TWR; FO APOLOGIZED FOR THE CONFUSION. ATC RESPONDED WITH; THERE WAS NO PROB. SUPPLEMENTAL INFO FROM ACN 340971: CHANGING HDG FROM 120 DEGS TO 070 DEGS WE WERE INSTRUCTED TO LEVEL OFF AT 7000 FT. WE ACKNOWLEDGED WITH 'ROGER 070' (MEANING HDG); AND DSNDED THROUGH 7000 FT AT 600-700 FT.