NMAC AT 1500 FT BTWN AN ARR B757 AND AN EXPERIMENTAL JET 5 MI E OF MYNN; FO.

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ZMA HANDED US OFF TO NASSAU APCH CTL APPROX 50 MI N OF ZQA VOR. APCH CTL WAS BUSY AND DID NOT ACKNOWLEDGE OUR CHK-IN UNTIL OUR THIRD OR FOURTH CALL. AFTER CHK-IN; WE CONTINUED OUR COURSE DIRECT TO THE ZQA VOR AT 13000 FT AND NORMAL SPD. APPROX 10 DME N OF THE ZQA VOR; APCH TOLD US TO FLY A HDG OF 140 DEGS FOR A VISUAL APCH TO RWY 23. WE ADVISED THEM THAT WE WOULD BE UNABLE HDG 140 DEGS DUE TO A SMALL TSTM JUST E OF THE ARPT; BUT WE COULD FLY A 170 DEG HDG. THIS WAS APPROVED AND WE FLEW DIRECTLY OVER THE MYNN ARPT. SHORTLY AFTERWARDS WE WERE GIVEN A DSCNT TO 7000 FT AND TOLD TO SLOW. WE MAINTAINED OUR 170 DEG HDG. APPROX 10-12 MI DUE S OF MYNN WE WERE TOLD TO DSND TO 3000 FT. APCH CTL WAS OBVIOUSLY VERY BUSY. AT THIS POINT; AN ACFT (UNKNOWN) ANNOUNCED THAT HE NEEDED A TURN BACK TO THE ARPT OR HE WOULD RUN OUT OF FUEL. APPROX 15 MI S OF MYNN; WE WERE GIVEN A L TURN TO 350 DEGS AND TOLD TO DSND TO 1500 FT MSL. THIS THEN PUT US ON AN INTERCEPT FOR THE VOR/DME 32 FINAL APCH COURSE. APPROX 10 MI SSE OF MYNN; WE WERE TOLD TO SLOW TO 150 KTS AND WE ADVISED THAT WE WOULD BE FOLLOWING A PIPER AZTEC (NOT IN SIGHT). AT THIS POINT WE HAD THE FIELD VISUALLY AND MAINTAINED OUR 350 DEG HDG AT 1500 FT MSL AND 150 KTS. WE WERE NEVER CLRED FOR THE VOR/DME 32 APCH OR A VISUAL APCH. I ATTEMPTED TO CONTACT APCH CTL AS WE PASSED THROUGH THE VOR/DME 32 FINAL APCH COURSE AND ALSO THE EXTENDED CTRLINE OF THE RWY; BUT COULD NEVER GET A WORD IN EDGEWISE. AT THIS POINT; AN ACFT (UNKNOWN) WAS TOLD TO TURN L HDG 140N DEGS. SEVERAL SECONDS LATER; AN ACFT (UNKNOWN) SAID 'IF I DO THAT; I'LL HIT HIM.' AT THAT POINT WE WERE APPROX 5 MI SSE OF MYNN WHEN WE GOT A TCASII TA. I LOOKED AT THE SCOPE AND GOT THE RELATIVE BEARING OF THE TARGET AND IMMEDIATELY ACQUIRED HIM VISUALLY. WE THEN GOT A TCASII RA. THE TIME FROM TCASII TA TO VISUAL ACQUISITION TO TCASII RA WAS APPROX 2-3 SECONDS. THE ACFT WAS A SMALL LOW WING 'EXPERIMENTAL JET' WITH ORANGE/RED ON THE WINGTIPS AND MAYBE ALSO ON THE FUSELAGE. HE PASSED APPROX 500 FT OFF THE L WINGTIP IN A L CLBING TURN WITH A 135 DEG XING ANGLE. I DID NOT REACT TO THE TCASII RA BECAUSE WE ACQUIRED HIM VISUALLY ALMOST IMMEDIATELY AND BASICALLY DIDN'T HAVE TIME TO REACT ANYWAY. WE FINALLY WERE ABLE TO COMMUNICATE TO APCH THAT WE HAD THE FIELD VISUALLY AND THEY THEN CLRED US FOR A VISUAL APCH. WE MADE A L TURN TO THE W AND INTERCEPTED THE RWY 32 EXTENDED CTRLINE AND LANDED. IN MY OPINION; NAS APCH CTL WAS TOTALLY OVERLOADED. THE CTLR APPEARED TO BE ON THE 'VERGE' OF LOSING CTL (SITUATIONAL AWARENESS) FROM THE TIME WE CHKED IN UNTIL THIS INCIDENT. CONTRIBUTING FACTORS WERE A VERY LOCALIZED TSTM E OF THE ARPT; A LOW FUEL STATE ACFT; AND THE 'RANDOM WANDERINGS OF THIS EXPERIMENTAL JET.'