A BE300 SUPER KING AIR; DSNDED BELOW ASSIGNED ALT AS IT APCHS TPA; FL.

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DSNDING FOR A VISUAL APCH TO RWY 36L AT TPA; WAS REACHING 4300 FT MSL WHEN ATC ADVISED TFC AT 12 O'CLOCK POS; 5 MI AT 4000 FT. I ADVISED FO TO CONFIRM WITH ATC THAT WE WERE CLRED TO 4000 FT (ALT ALERTER HAD BEEN SET FOR 4000 FT AND FO CONFIRMED HE THOUGHT WE HAD BEEN CLRED TO 4000 FT). ATC ADVISED WE WERE CLRED TO 5000 FT. WE CLB IMMEDIATELY BACK TO 5000 FT. WHEN ON GND CALLED TRACON AND TALK TO WATCH SUPVR. I STATED WE HAD THOUGHT WE WERE CLRED TO 4000 FT AND THE FO SET ALT ALERTER TO 4000 FT. HE CONFIRMED TAPE VERIFIED OUR CLRNC TO 5000 FT AND WE ACKNOWLEDGED 5000 FT. I HAD BEEN LISTENING TO ATIS WHEN THE CLRNC WAS GIVEN AND ASSUMED THE FO WAS CORRECTLY SETTING THE ASSIGNED ALT. WE BROKE THE CHAIN OF CHKS WHEN I WAS LISTENING TO ATIS AND THE FO WAS TALKING TO ATC. THIS NEGATED A REASON FOR HAVING 2 PLTS. I LEARNED THAT IN THE FUTURE DON'T ASSUME THE OTHER PLT IS CORRECTLY PROVIDING CRITICAL FLT INFO AND CONFIRM ASSIGNED ALTS WHEN YOU DID NOT ACTUALLY HEAR THEM ASSIGNED. SUPPLEMENTAL INFO FROM ACN 381960: DURING LETDOWN WAS CLRED TO DSND TO 5000 FT BUT ERRONEOUSLY ENTERED 4000 FT IN ALT ALERTER. CAPT DSNDED TO 4300 FT WHEN ADVISORY OF TFC AT 4000 FT CAUSED CREW TO QUESTION OUR ASSIGNED ALT. QUESTIONED ATC WHICH ADVISED 5000 FT. ACFT IMMEDIATELY WAS CLBED FROM 4300 FT TO 5000 FT. MUST CONSTANTLY GUARD AGAINST HUMAN ERROR AND CREW MUST XCHK EACH OTHER.