Let $\mathcal{A}$ be a subcategory of $\mathcal{C}$. Let $D(\mathcal{A})$ and $D(\mathcal{C})$ be the associated derived categories. We can define $D_\mathcal{A}(\mathcal{C}) = \{X \in \mathcal{C}\hspace{2pt} |\hspace{2pt} H^i(X) \in \mathcal{A}\}$. We have natural inclusions $$D(\mathcal{A}) \to D_\mathcal{A}(\mathcal{C}) \to D(\mathcal{C})$$

My question is this: When is $D(\mathcal{A}) \to D_\mathcal{A}(\mathcal{C})$ an equivalence of categories?

I have read that it is not always an equivalence, but that it often is. I have examples of when it is: Is there an easy example for when it isn't?