In this paper we describe an experimental technique for computing realistic values of the parameter--uniform order of convergence and error constant in the maximum norm associated with a parameter--uniform numerical method for solving singularly perturbed problems. We employ the technique to compute Reynolds--uniform error bounds in the maximum norm for the numerical solutions generated by a fitted--mesh upwind finite difference method applied to Prandtl's problem arising from laminar flow past a thin flat plate. Thus we illustrate the efficiency of the technique for finding realistic parameter--uniform error bounds in the maximum norm for the approximate solutions generated by numerical methods for which no theoretical error analysis is available.