In short: No.
Y? T-mobile launched its 3G/4G on UMTS: Band I (2100);UMTS: Band IV (1700/2100) and soon to be 1900.
Digicel/Lime/Claro and AT&T uses the 850MHz and maybe the 1900Mhz for their 3G/4G. To ensure your phone gets 3G in Jamaica, ensure you phone supports the 850MHz UMTS band.
However some non-blackberry T-Mobile phones now carry the 850Mhz band such as the HTC Amaze, SGS II (Hercules).