In this paper, we study the robustness characteristics of a recently developed concurrent learning model reference adaptive control approach to time-varying disturbances and system uncertainties. Specifically, the commonly-used constant (or slowly time-varying) assumption on disturbances and system uncertainties for this particular adaptive control approach is replaced with its bounded counterpart with piecewise continuous and bounded derivatives. Based on the Lyapunov's direct method, we then show that the solutions of the closed-loop system are uniformly ultimately bounded, without requiring a modification term in the adaptive law. Estimates for the ultimate bound and the exponential convergence rate to that ultimate bound are further provided. According to these estimates and illustrative numerical examples, similarities and differences between concurrent learning and one of the well-known robustness modifications in adaptive control, namely sigma modification, are explored.