Semiquantitative tests are considered of models, proposed earlier by this author, in which highTcsuperconductivity follows from the exchange of unspecified intermediate bosons (IB) with &ohgr;≫&ohgr;phononbetween fermionic pairs in different CuO2layers. Earlier predictions of the dependence ofTc(n) on the numbernof layers per cluster are extended to include the possibility thatTc(n=1)<∼10 K in the Tl 1201 and 2201 compounds. In common with other authors rapid saturation ofTc(n) with increasingnis found here. Nonetheless, the experimental value ofTc(n=1) in the Tl and Bi systems can play an important role in discriminating between the various models empirically.