ACR X NON ADHERENCE TO ATC CLRNC FAILED TO MEET ALT RESTRICTION.

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I OPERATED A FLT X FROM CANCUN MEXICO TO DETROIT; MI. THIS WAS A PUBLIC CHARTER FLT. ACFT TYPE WAS AN L-1011. ON TAXI OUT FROM CUN; OUR INITIAL CLRNC READ: CANCUN FRISH MINOW CIGAR FLT PLAN RTE CLB TO FL290. UPON HDOF FROM CANCUN TO MERIDA CTR WE WERE RECLRED TO FL370 WITH A RESTRICTION TO CROSS THE FIR BOUNDARY LEVEL AT FL370. WE ACCEPTED THE CLRNC AND CONTINUED THE CLB. BECAUSE OF TEMPS ALOFT BEING SEVERAL DEGS WARMER THAN FORECAST; IT BECAME APPARENT TO US THAT WE WOULD BE UNABLE TO MEET THE FIR BOUNDARY XING RESTRICTION. WHEN AT A POS OF 10 MI FROM THE BOUNDARY WE ADVISED MERIDA CTR THAT WE WERE CLBING THROUGH FL350 AND WOULD BE UNABLE TO CROSS THE FIR AT FL370. MERIDA CTR ACKNOWLEDGED AND INSTRUCTED US TO CONTACT ZMA AT 86 DEGS W LONGITUDE ON FREQ 133.9. STILL CLBING TO FL370; WE RELAYED A PROGRESS RPT TO NY ARINC ON FREQ 130.7. THAT POS RPT WAS AS FOLLOWS: þCHK ABEAM 'FRISH' AT TIME; CLBING TO FL370 ESTIMATE 'MINOW' TIME; CIGAR NEXT.' AND INCLUDED A COMMENT ABOUT BEING OFFSET UP TO 20 NM W OF COURSE FOR WX. AT 86 DEGS W WE CALLED ZMA AND RELAYED OUR PROGRESS RPT TO THEM. AT THE TIME OF INITIAL CONTACT; WE WERE STILL CLBING; PASSING FL365 FOR FL370. ZMA ADVISED OF OPPOSITE DIRECTION TFC AT FL350 WHICH SHORTLY APPEARED ON OUR TCASII DISPLAY AT 30-35 NM RANGE. WE SELECTED A HIGHER VERT SPD; TRADED SOME AIRSPD FOR ALT; AND ZMA ADVISED US THAT THEY THOUGHT WE WERE ALREADY LEVEL AT FL370; BEING ADVISED OF OUR EARLIER XING RESTRICTION AT THE FIR. ALSO; ARINC HAD PASSED ALONG OUR PROGRESS RPT TO ZMA; BUT THEY INDICATED THAT WE HAD CHKED FRISH LEVEL AT FL370 OF INSTEAD OF CLBING TO FL370. IN FACT; AT THE TIME WE PASSED FRISH; WE WERE PASSING FL315 IN THE CLB. MERIDA CTR NEVER PASSED ALONG THE INFO; APPARENTLY; THAT WE WERE UNABLE TO REACH FL370 BY THE FIR BOUNDARY. BUT NEITHER DID THEY AMEND OUR CLRNC OR EXPRESS ANY CONCERN WHEN WE ADVISED THEM THAT WE'D NOT BE LEVEL AT FL370 AT THE FIR BOUNDARY. AS IS OFTEN THE CASE; COM BROKE DOWN. SUBSEQUENTLY; I WAS REQUESTED TO CONTACT ZMA BY PHONE. IN MY CONVERSATION WITH THE AREA MGR IN CHARGE (AMIC) I SHARED WITH HIM MY SIDE OF THE STORY. HE EXPLAINED THE DIFFICULTY WITH COMMUNICATING WITH MERIDA; THOUGH THEY SHARE A LARGE BOUNDARY. COM IS ALL RELAYED THROUGH ZHU AND ARINC. THEY SPECIFICALLY SAID THAT CONFLICTS SUCH AS THIS OCCUR FREQUENTLY; INCLUDING INCIDENTS THAT REQUIRE EVASIVE MANEUVERING. WITH THE TFC VOLUME THAT EXISTS OVER THE GULF; THIS SEEMS AN UNWISE SYS FOR COM AND COORD. ZMA'S CONCERN IS THE LOSS OF NON RADAR SEPARATION; THE PRINCIPAL CAUSE OF WHICH SEEMS TO BE POOR COM BTWN MERIDA AND ZMA. ALSO OF CONCERN TO ME; THE APPARENT NEED FOR ALL COM TO BE RELAYED BY A THIRD PARTY.