FLC OF A CANADAIR CL65 REGIONAL JET OVERSHOT ALT DURING CLB DUE TO MISUNDERSTANDING OF THE CLRNC ALT AND NOT NOTICED BY THE ARTCC CTLR DURING THE CAPT'S READBACK.

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THE NASSAU TWR AND DEP FREQS WERE VERY BUSY RESULTING IN US FIRST BEING LEFT ON RWY HDG (136 DEGS) UP TO 4000 FT AND THEN ON A 230 DEG UP TO 12000 FT FOR APPROX 20 MI. I; THE CAPT AND PNF; ATTEMPTED REPEATEDLY TO OBTAIN AN ON COURSE HDG FROM DEP. FINALLY SUCCESSFUL; WE WERE ASSIGNED A 360 DEG HDG TO INTERCEPT BR22V. WHEN WE SWITCHED TO ZMA; WE RECEIVED A CLRNC TO PROCEED DIRECT TO FLL VOR AND A CLB WHICH I READ BACK AS FL220. A DISCUSSION THEN ENSUED WITH THE ZMA CTLR OVER NASSAU LEAVING US ON THE 230 DEG HDG FOR SO LONG. THE ZMA CTLR SAID THAT HE SAW US ON HIS RADAR AND WONDERED WHAT WAS GOING ON; SO HE HAD PLACED A CALL TO NASSAU TO REMIND THEM ABOUT US. I RESPONDED THAT WE WERE GRATEFUL BECAUSE IT WAS VERY BUSY AND THE NASSAU CTLR HAD APPARENTLY FORGOTTEN ABOUT US. LATER IN THE CLB THROUGH APPROX FL180; WE WERE TOLD THAT THERE WAS OPPOSITE DIRECTION TFC AT FL230 AND TO EXPECT HIGHER PASSING. APCHING FL200 WE WERE SWITCHED TO A NEW MIAMI FREQ. A BRIEF PERIOD OF TIME PASSED BEFORE THERE WAS A BREAK ON THE FREQ ENABLING ME TO CHK ON. AS WE PASSED THROUGH FL205 WE RECEIVED A TA; SO I SELECTED THE TFC DISPLAY AND SAW TFC 500 FT ABOVE APCHING; AND THEN PASSING 5 MI OFF OUR L. SHORTLY THEREAFTER THE CTLR ACKNOWLEDGED OUR CHK ON; INFORMED US OF TFC PASSING OFF OUR L AND ASKED WHAT ALT WE WERE ASSIGNED. I RESPONDED FL220 TO WHICH HE SAID THAT HE SHOWED FL200 AND THAT THEY WOULD BE PULLING TAPES. IN HINDSIGHT; I BELIEVE THAT THE DELAYS ENCOUNTERED WITH THE BAHAMIAN CTLR AND THE ENSUING DISCUSSION WITH THE ZMA CTLR OVER OUR TREATMENT; RESULTED IN AN INATTN TO DETAIL REGARDING THE ALT ASSIGNMENT AND READ BACK ON BOTH THE CTLR'S AND MY PART.