This paper studies the delay robustness of an L-1 adaptive controller designed for systems with unknown matched nonlinearities and unknown input-gain matrices. The analysis establishes rigorously the existence of a positive lower bound for the closed-loop system's time-delay margin (TDM), provided that a filter bandwidth and an adaptive gain are chosen sufficiently large. In this case, if the input delay is below a critical value, then the state and control input of the control system follow those of a nonadaptive, robust reference system closely. The analysis also suggests a way to estimate this lower bound for the delay robustness using Pade approximants. Results fromforward simulation are consistent with the Pade estimate and with an explicit upper bound on the TDM which decays to 0 as the filter bandwidth grows without bound.