  The quantum master equation is a widespread approach to describing open
quantum system dynamics. In this approach, the effect of the environment on the
system evolution is entirely captured by the dynamical generator, providing a
compact and versatile description. However, care needs to be taken when several
noise processes act simultaneously or the Hamiltonian evolution of the system
is modified. Here, we show that generators can be added at the master equation
level without compromising physicality only under restrictive conditions.
Moreover, even when adding generators results in legitimate dynamics, this does
not generally correspond to the true evolution of the system. We establish a
general condition under which direct addition of dynamical generators is
justified, showing that it is ensured under weak coupling and for settings
where the free system Hamiltonian and all system-environment interactions
commute. In all other cases, we demonstrate by counterexamples that the exact
evolution derived microscopically cannot be guaranteed to coincide with the
dynamics naively obtained by adding the generators.
