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We present the results of detailed transient photoluminescence measurements on the isoelectronic bound excitons associated with the deep acceptors In and T1 in Si. The novel results are explained in terms of a model in which the binding centers can exist in more than one configuration, and in which the bound excitons can make transitions from one configuration to the other. Although the nature of the binding center and the exciton states is not yet known, the transient results are well described by rate equations based upon this model. Additional support is provided by a study of the effects of excitation pulse length on the Si: In transient results, as well as by comparing above-gap excitation versus resonant excitation of the bound excitons.