Is it consistent (with ZFC) that there is a cardinal $\kappa$ and $m_1$ and $m_2$ two measures on $\kappa$, both $\kappa$-additive, such that
- $m_1$ is atomless, and
- $m_2$ has an atom?
In that case, the existence of $m_1$ implies that the continuum is at least as large as $\kappa$ and the existence of $m_2$ implies that $\kappa$ is inaccessible. That would mean a radical failure of CH. Of course, we know from Easton's Theorem that the size of the continuum can be "almost" anything. Even then, I am curious to know if $m_1$ and $m_2$ can coexist.
I assume you want the measures to be $0$ on singletons. If this is the intention, it is impossible for both measures to coexist, for the reason that you identify: $m_1$ would force $\kappa$ to be (atomlessly) real-valued measurable, which implies that $\kappa\le\mathfrak c=|\mathbb R|$, while $m_2$ would force $\kappa$ to be a measurable cardinal, and therefore strictly larger than $\mathfrak c$.
(Of course, if you do not add the requirement that the measures vanish on singletons, then we have a silly example by starting with a real-valued measurable cardinal and a witnessing measure $m_1$, and letting $m_2$ be the $\{0,1\}$-measure corresponding to a principal ultrafilter on $\kappa$.)
Kanamori's book on large cardinals and Fremlin's survey on real-valued measurability are both good places to read on this topic. Both discuss in an accessible manner the details required to fully understand the first paragraph above.