Consider bipartite (qubit) systems. The classical mutual information between a pair of binary registers, $$I(X:Y)\equiv H(X)+H(Y)-H(X,Y),$$ is always lesser than $1$ (and non-negative). On the other hand, the quantum mutual information of a bipartite state $\rho$, defined as $$I(\rho) \equiv S(\rho_A) + S(\rho_B) - S(\rho),$$ with $S(\rho)$ von Neumann entropy of $\rho$, and $\rho_A\equiv\operatorname{Tr}_B(\rho)$, $\rho_B\equiv\operatorname{Tr}_A(\rho)$, satisfies $0\le I(\rho)\le 2$.
If $\rho$ is pure, we also know that $I(\rho)=2J(\rho)$, where $J(\rho)$ is the accessible mutual information (the one obtained computing the mutual information via the conditional entropy, maximising over the possible measurement choices). Therefore, a pure $\rho$ is separable (i.e. a product state) iff $I(\rho)=J(\rho) = 0$.
For a more general classical-quantum state, some $\rho=\sum_i p_i \,|i\rangle\!\langle i|\otimes \rho_i$, we have $$I(\rho) = H(\mathbf p)+S\left(\sum_{i}p_{i}\rho_{i}\right)-S(\rho), \\ S(\rho) = \sum_i p_i S(\rho_i) + H(\mathbf p),$$ and thus $$ I(\rho) = S\left(\sum_{i}p_{i}\rho_{i}\right) - \sum_i p_i S(\rho_i) \le S\left(\sum_{i}p_{i}\rho_{i}\right) \le 1, $$ because $S(\sigma)\le1$ for any single-qubit state $\sigma$.
These are all examples of separable states with quantum mutual information $I(\rho)\le 1$. More generally, it's clear that a state can give $I(\rho)>1$ only if it has nonzero discord, so a separable state with $I(\rho)>1$ would have to be discordant. But classical-quantum states are separable and can have nonzero discord (with respect to measurements on the second space), but still always give $I(\rho)\le1$. Other classical examples of discordant separable states that are not classical-quantum are Werner states: $$\rho_z = \frac{1-z}{4}I +z |\Phi^+\rangle\!\langle\Phi^+| =\frac14\begin{pmatrix}1+z & 0&0& 2z \\ 0& 1-z & 0 & 0 \\ 0&0&1-z&0 \\ 2z & 0 & 0 & 1+z\end{pmatrix}, \\ |\Phi^+\rangle\equiv\frac{1}{\sqrt2}(|00\rangle+|11\rangle).$$ These are separable for $z\le1/3$, but as discussed in (Ollivier, Zurek 2001), have nonzero discord. Their quantum mutual information reads $$I(\rho_z) = \frac14\left[ (1+3z) \log_2(1+3z) + 3(1-z)\log_2(1-z) \right],$$ which is smaller than $1$ for $z\le 1/3$.
Is the above a general feature? In other words, do all separable states have $I(\rho)\le1$?