Standard necessary conditions for optimal control problems with pathwise state constraints supply no useful information about minimizers in a number of cases of interest, e.g., when the left endpoint of state trajectories is fixed at x(0) and x(0) lies in the boundary of the state constraint set; in these cases a nonzero, but nevertheless trivial, set of multipliers exists. We give conditions for the existence of nontrivial multipliers. A feature of these conditions is that they allow nonconvex velocity sets and measurably time-dependent data. The proof techniques are based on refined estimates of the distance of a given state trajectory from the set of state trajectories satisfying the state constraint, originating in the dynamic programming literature.