Yes, those are accurate statements.
The inclusion $\mathbb{Z}\hookrightarrow \mathbb{Q}$ is both a monomorphism and epimorphism in the category $\mathsf{Ring}$ (rings and ring homomorphisms), but not an isomorphism. The inclusion $\mathbb{Q}\hookrightarrow\mathbb{R}$ is both a monomorphism and epimorphism in the category $\mathsf{Haus}$ (Hausdorff topological spaces and continuous maps), but it is not an isomorphism.
In the category $\mathsf{Set}$, monomorphisms and epimorphisms are precisely the injective and surjective maps, respectively, so that a map of sets that is both a monomorphism and epimorphism is a bijection, i.e. an isomorphism of sets.
More examples and information can be found in Wikipedia and in Mac Lane's Categories for the Working Mathematician.