The transcriptional network determines a cell's internal state by regulating protein expression in response to changes in the local environment. Due to the interconnected nature of this network, information encoded in the abundance of various proteins will often propagate across chains of noisy intermediate signaling events. The data-processing inequality (DPI) leads us to expect that this intracellular game of "telephone" should degrade this type of signal, with longer chains losing successively more information to noise. However, a previous modeling effort predicted that because the steps of these signaling cascades do not truly represent independent stages of data processing, the limits of the DPI could seemingly be surpassed, and the amount of transmitted information could actually increase with chain length. What that work did not examine was whether this regime of growing information transmission was attainable by a signaling system constrained by the mechanistic details of more complex protein-binding kinetics. Here we address this knowledge gap through the lens of information theory by examining a model that explicitly accounts for the binding of each transcription factor to DNA. We analyze this model by comparing stochastic simulations of the fully nonlinear kinetics to simulations constrained by the linear response approximations that displayed a regime of growing information. Our simulations show that even when molecular binding is considered, there remains a regime wherein the transmitted information can grow with cascade length, but ends after a critical number of links determined by the kinetic parameter values. This inflection point marks where correlations decay in response to an oversaturation of binding sites, screening informative transcription factor fluctuations from further propagation down the chain where they eventually become indistinguishable from the surrounding levels of noise.