In this article, we highlight a common misinterpretation that can arise when reading the physics literature on the conservation of adiabatic invariants under slow parameter changes in classical integrable systems with multiple degrees of freedom (N > 1). As illustrative examples, we discuss the classical anisotropic oscillator and the Foucault pendulum, which demonstrate the non-conservation of action variables in integrable classical mechanical systems undergoing adiabatically slow evolution.