ALT DEV ALT OVERSHOT ALT EXCURSION.

Narrative

WE WERE FLYING FROM CANCUN; MEXICO TO TAMPA; FL; ON UB646 WITH THE ACFT LEVEL AT FL250. WE WERE UNDER THE CTL OF HAVANA CENTER; BUT NOT IN RADAR CONTACT YET. AT APPROX ROBIN INTXN; HAVANA GAVE US A CLRNC TO CLB. THE ALT I THOUGHT I HEARD WAS FL370. THE COPLT READ BACK TO HAVANA FL370 AND SET 37000 IN THE ALT WARNING INDICATOR. THE CAPT ACKNOWLEDGED FL370 IAW OUR COMPANY PROCS AND STARTED TO CLB THE ACFT. HAVANA CENTER DID NOT QUESTION THE COPLTS READBACK AND NO ONE IN OUR COCKPIT RAISED THE ISSUE OF POSSIBLY 'HEARING AN ALT OTHER THAN FL370. WHEN OUR ACFT WAS CLBING THROUGH ABOUT 30500 FT MSL THE COPLT MENTIONED THAT HIS TFC ALERT AND COLLISION AVOIDANCE SYS (TCAS) INDICATOR SHOWED TFC ABOVE US AND JUST OF OUR R WING. WE LOOKED IN THIS DIRECTION AND SAW ANOTHER ACR LGT. THE CAPT STOPPED THE CLB AT FL310. THE COPLT CALLED ACFT Y TO VERIFY THEIR ALT. THEY SAID THEY WERE LEVEL AT FL330. AT THIS POINT THE COPLT CALLED HAVANA CENTER TO ASK ABOUT OUR ALT CLRNC. HAVANA RESPONDED THAT WE WERE CLRED TO FL270. WE TOLD HAVANA WE WERE LEVEL AT FL310. AT THIS POINT HAVANA SAID IT WAS 'OK' TO STAY AT FL310. A SHORT TIME LATER; HAVANA ESTABLISHED RADAR CONTACT AND GAVE US SEPARATION VECTORS BEFORE HANDING US OFF TO MIAMI CENTER. A CONTRIBUTING FACTOR IN THIS INCIDENT WAS THAT OUR FLT PLAN SHOWED FL370 AS OUR FINAL ALT. SO; WHEN HAVANA ISSUED OUR CLRNC TO CLB; WE WERE EXPECTING FL370. ANOTHER FACTOR MAY HAVE BEEN THE LANGUAGE BARRIER/DIFFERENCE INVOLVED. WHILE THE HAVANA CTLRS ENGLISH WAS VERY GOOD; WE DID NOT HEAR FL270. ALSO; EVEN THOUGH OUR COPLT MADE A CLR; DISTINCT READBACK OF THE CLRNC; THE CTLR DID NOT HEAR FL370.