ATX X HAD NMAC LTSS FROM ATX Y. SYS ERROR. SIMILAR CALL SIGNS.

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ATX X WAS INBOUND FROM THE W DSNDING FROM 7000 TO 3000. ATX Y WAS INBOUND FROM THE E AT 4000 S OF SJU HDG 280. ATX X WAS VECTORED S HDG 180 FOR SEQUENCE. ATX Y WAS VECTORED N HDG 010. ATX X WAS VECTORED NE TO HDG 060 LEVEL AT 3000. THE ACFT WERE STILL APPROX 10 MI APART AND 1000 FT APART. MY SUPVR WAS WATCHING THE SESSION AND POINTED OUT A POSSIBLE CONFLICTION AND I ADVISED HIM I WAS AWARE OF IT. I PLANNED TO TURN BOTH ACFT TO PARALLEL HDGS AND TURN THEM ON TO THE LOC FOR RWY 10. I ISSUED ATX Y A TURN TO HDG 330 AND DSCNT TO 1500. HE ASKED ME TO REPEAT THE ALT AND I ISSUED HIM THE ENTIRE CLRNC AGAIN. HE 'ROGERED' THE XMISSION. I THEN WENT TO ATX X AND ISSUED A TURN TO HDG 330 AND DSCNT TO 1500. I RECEIVED A REPLY TO THAT XMISSION AND CONTINUED TO WORK THE OTHER ACFT ON FREQ. WHEN I CHKED THEIR POS AGAIN THEY WERE CONVERGING AND I ISSUED ATX X TO TURN 10 DEGS FURTHER L AND MAINTAIN ALT AND EXPEDITE HIS TURN. I INSTRUCTED ATX Y TO EXPEDITE HIS DSCNT. HE INFORMED ME HE HAD THE COMPANY IN SIGHT AND I INSTRUCTED HIM TO MAINTAIN VISUAL SEPARATION. ATX X WAS STILL AT 3000. I ADVISED ATX X THAT I NEEDED HIM ON HIS ASSIGNED HDG AND HE STATED THAT HE WAS FLYING HDG 050. AFTER REVIEWING THE TAPE WHEN I HAD WANTED TO TURN ATX X TO A PARALLEL COURSE I USED THE ATX COMPANY WRONG A/N. WRONG ACFT REPLIED AND SINCE I HAD MADE 2 XMISSIONS TO HIM JUST PRIOR; I 'HEARD' WHAT I WANTED TO HEAR. CONSEQUENTLY; ATX X WAS STILL ON A HDG OF 060 AND WRONG ACFT WAS HDG 330 AND DSNDING. THE ATX COMPANY HERE USES FAR TOO MANY FLTS WITH SIMILAR SOUNDING CALLS IN THE AIR AT THE SAME TIME. THIS HAS CAUSED PROBLEMS WITH OTHER CTLRS HERE BUT SO FAR NOT OF THIS MAGNITUDE.