Computer simulations of photon transport through an absorbing and/or scattering medium form an important research tool in astrophysics. Nearly all software codes performing such simulations for three-dimensional geometries employ the Monte Carlo (MC) radiative transfer (RT) method, including various forms of biasing to accelerate the calculations. Because of the probabilistic nature of the MC technique, the outputs are inherently noisy, but it is often assumed that the average values provide the physically correct result. We show that this assumption is not always justified. Specifically, we study the intensity of radiation penetrating an infinite, uniform slab of material that absorbs and scatters the radiation with equal probability. The basic MCRT method, without any biasing mechanisms, starts to break down for transverse optical depths tau greater than or similar to 20 because so few of the simulatedphoton packets reach the other side of the slab. When including biasing techniques such as absorption/scattering splitting and path length stretching, the simulated photon packets do reach the other side of the slab but the biased weights do not necessarily add up to the correct solution. While the noise levels seem to be acceptable, the average values sometimes severely underestimate the correct solution. Detecting these anomalies requires the judicious application of statistical tests, similar to those used in the field of nuclear particle transport, possibly in combination with convergence tests employing consecutively larger numbers of photon packets. In any case, for transverse optical depths tau greater than or similar to 75 the MC methods used in our study fail to solve the one-dimensional slab problem, implying the need for approximations such as a modified random walk.