We present a model for analyzing the reliability of component-based software systems. Each software component is described as a finite state machine whose transitions have failure probabilities that reflect the unreliable execution of elementary component operations. Alternatively, we model a component as a finite state machine with an associated reliability figure that expresses the overall reliability of the component. Using our model, we prove that there is no algorithm that can accurately calculate the reliability of a system of finite state machines with known or estimated overall reliabilities or transition failure probabilities. The same negative result holds even if we only want to approximate the system reliability.