In this work we study isolated flexible interaction-site chain molecules comprised of n square-well-sphere monomers. We compute the "exact" density of states of such chains with n=3, 4, 5, and 6 and use these results to carry out a detailed study of the thermodynamics of chain collapse. We find that for a certain range of square-well diameters both the n=5 and 6 chains undergo a multistage collapse transition with decreasing temperature (as indicated by multiple peaks in the specific heat and, in some cases, by discrete steps in the internal energy). The collapse behavior is strongly affected by the presence of a set of "cut-off" values of the square-well diameter at which the ground-state energy of the chain undergoes a discontinuous change. For a well diameter slightly larger than any one of these cut-off values, the chain initially collapses to an energy state above the ground state and then undergoes a separate low temperature transition to the ground state. There is a large specific heat peak associated with this latter transition which grows in height as the cut-off diameter is approached and abruptly vanishes at the cut-off diameter itself. Thus, the collapse behavior of short square-well chains is not a smooth function of well diameter. We discuss the implications of this finding for the use of the square-well chain model to study the polymer collapse transition. (C) 2003 American Institute of Physics.