Are two languages regular if the union is regular?
Pick $\Sigma = \{a,b\}$, $L_1 = \{ a^n b^n : n \in \mathbb{N}\}$ and $L_2 = \Sigma^* \setminus L_1$.

The union $L_1 \cup L_2$ is $\Sigma^*$ and hence is regular, but none of $L_1$ and $L_2$ is.