In this paper fitted finite difference methods on a uniform mesh with internodal spacing h, are considered for a singularly perturbed semilinear two-point boundary value problem. It is proved that a scheme of this type with a frozen fitting factor cannot converge epsilon-uniformly in the maximum norm to the solution of the differential equation as the mesh spacing h goes to zero. Numerical experiments are presented which show that the same result is true, for a number of schemes with variable fitting factors.