Directed self-assembly of designed viral capsids holds significant potential for applications in materials science and medicine. However, the complexity of preparing these systems for assembly and the difficulty of quantitative experimental measurements on the assembly process have limited access to critical mechanistic questions that dictate the final product yields and isomorphic forms. Molecular simulations provide a means of elucidating self-assembly of viral proteins into icosahedral capsids and are the focus of the present study. Using geometrically realistic coarse-grained models with specialized molecular dynamics methods, we delineate conditions of temperature and coat protein concentration that lead to the spontaneous self-assembly of T = 1 and T = 3 icosahedral capsids. In addition to the primary product of icosahedral capsids, we observe a ubiquitous presence of nonicosahedral yet highly symmetric and enclosed aberrant capsules in both T = 1 and T = 3 systems. This polymorphism in assembly products recapitulates the scope and morphology of particle types that have been observed in mis-assembly experiments of virus capsids. Moreover, we find that this structural polymorphism in the end point structures is an inherent property of the coat proteins and arises from condition-dependent kinetic mechanisms that are independent of the elemental mechanisms of capsid growth (as long as the building blocks of the coat proteins are all monomeric, dimeric, or trimeric) and the capsid T number. The kinetic mechanisms responsible for self-assembly of icosahedral capsids and aberrant capsules are deciphered; the self-assembly of icosahedral capsids requires a high level of assembly fidelity, whereas self-assembly of nonicosahedral capsules is a consequence of an off-pathway mechanism that is prevalent under nonoptimal conditions of temperature or protein concentration during assembly. The latter case involves kinetically trapped dislocations of pentamer-templated proteins with hexameric organization. These findings provide insights into the complex processes that govern viral capsid assembly and suggest some features of the assembly process that can be exploited to control the assembly of icosahedral capsids and nonicosahedral capsules.