The observability of the attitude and angular rate estimation problem is analyzed using notions from nonlinear systems theory. Exploiting the distinctive structure of the system, expressions for its Lie derivatives of any order are derived. The observability mapping is then constructed and analyzed, yielding that the system is non-uniformly observable. Necessary and sufficient conditions for the system to become unobservable are derived. It is further shown that the angular rate unobservability conditions, thus derived, are a generalization of previously obtained results. Additionally, the observability of the inertia tensor is examined, and it is shown that the inertia is non-uniformly observable. A sufficient condition is given under which the momentum wheel control input renders the inertia unobservable. This phenomenon is numerically demonstrated using a Bayesian grid-based filter that is applied to the estimation of the attitude and angular rate of a spacecraft subjected to inertia tensor uncertainty.