It has been observed that a Newton-GMRES method globalized by backtracking (linesearch, damping) may be less robust when high accuracy is required of each GMRES solve than when less accuracy is required. We offer a possible explanation for this phenomenon, together with an illustrative numerical experiment. These indicate that requiring high accuracy may give rise to a long step, especially if the Jacobian has small singular values; moreover, the step may be nearly orthogonal to the gradient of the nonlinear residual norm, especially if the Jacobian is ill-conditioned. Such a step may require so many reductions in length for acceptability that the backtracking routine may declare failure before an acceptable step is found. This work is joint with John N. Shadid and Raymond S. Tuminaro, Sandia National Laboratories.