When is a pullback also a pushout?
$\require{amsCD}$ Yes. The most familiar setting where this happens is in abelian categories, where the commutative square
$$\begin{CD} A @>f_b>> B\\ @VVf_cV @VVg_bV\\ C @>g_c>> D \end{CD}$$
is a pullback square iff the corresponding sequence
$$0 \to A \xrightarrow{f_b \oplus f_c} B \oplus C \xrightarrow{g_c - g_b} D$$
is exact. Similarly, the square is a pushout square iff the sequence
$$A \xrightarrow{f_b \oplus f_c} B \oplus C \xrightarrow{g_c - g_b} D \to 0$$
is exact. Hence the square is both a pullback and a pushout square iff the sequence
$$0 \to A \xrightarrow{f_b \oplus f_c} B \oplus C \xrightarrow{g_c - g_b} D \to 0$$
is exact.
$\require{amscd}$
There are some special cases of this phenomenon where things can be said. Consider the following diagram:
$$\begin{CD} A @>f>> B\\ @VVgV @VVhV\\ C @>k>> D \end{CD}$$
Suppose that $f$ and $k$ are monomorphisms. Then in many categories, the implication (pushout $\Rightarrow$ pullback) holds. This holds in any topos, in any abelian category, and more generally in any adhesive category. According to Stefan Hamcke's comment, this holds in $k$-spaces. But we typically don't get the converse implication (pullback $\Rightarrow$ pushout). For example, in $\mathsf{Set}$, let $B,C$ be any subsets of $D$ such that $B \cup C \subsetneq D$, and let $A = B \cap C$. Similar counterexamples work in $\mathsf{Ab}$.
Suppose that $f$ and $k$ are monomorphisms and $g$ and $h$ are epimorphisms. The facts Qiaochu discusses can be used to show that in an abelian category, the equivalence (pushout $\Leftrightarrow$ pullback) holds. In $\mathsf{Set}$, though, the leftward implication fails -- e.g. let $A = C = \emptyset$ and $B = 2$, $D=1$.
Suppose that $g$ and $h$ are epimorphisms. Then in any abelian category, the dual of (1) tells us that the implication (pullback $\Rightarrow$ pushout) holds. But this is not the case in $\mathsf{Set}$.
As an special case of (3), suppose that $f,g,h,k$ are all epimorphisms. If I'm not mistaken, we get the implication (pullback $\Rightarrow$ pushout) in $\mathsf{Set}$ and so also in any topos. But the converse implication rarely holds.
Abelian categories and topoi are about as nice as it gets, so I would expect that in less nice categories, the best one can do is similar implications with more restrictive assumptions on the morphisms involved.