In this paper, we explore both numerically and analytically the robustness of a synchronous Boolean network governed by rule 126 of cellular automata. In particular, we explore whether or not the introduction of noise into the system has any discernable effect on the evolution of the system. This noise is introduced by changing the states of a given number of nodes in the system according to certain rules. New mathematical models are developed for this purpose. We use MATLAB to run the numerical simulations including iterations of the real system and the model, computation of Lyapunov exponents, and generation of bifurcation diagrams. We provide a more in-depth fixed-point analysis through analytic computations paired with a focus on bifurcations and delay plots to identify the possible attractors. We show that it is possible either to attenuate or to suppress entirely chaos through the introduction of noise and that the perturbed system may exhibit very different long-term behavior than that of the unperturbed system.