When does Cantor-Bernstein hold?
Whenever the objects in your category can be classified by a bounded collection of cardinal invariants, then you should expect to have the Schroeder-Bernstein property.
For example, vector spaces (over some fixed field $K$) or algebraically closed fields (of some fixed characteristic) can each be classified by a single cardinal invariant: the dimension of the vector space, or the transcendence degree of the field.
More interesting example: countable abelian torsion groups. Suppose A and B are two such groups, $A$ is a direct summand of $B$, and vice-versa; are they isomorphic? By Ulm's Theorem, $A$ and $B$ are determined up to isomorphism by countable sequences of cardinal numbers -- namely, the number of summands of $\mathbb{Z}_p^\infty$ and the "Ulm invariants," which are dimensions of some vector spaces associated with $A$ and $B$. All of these invariants behave nicely with respect to direct sum decompositions, so it follows that $A$ and $B$ are isomorphic. (See Kaplansky's Infinite Abelian Groups for a very nice, and elementary, proof of all this.)
If you like model theory, I could tell you a lot about when the categories of models of a complete theory have the Schroeder-Bernstein property (under elementary embeddings). If not, at least I can tell you this:
Categories of structures with "definable" partial orderings with infinite chains (e.g. real-closed fields, atomless Boolean algebras) will NOT have the S-B property. Again, I need some model theory to make this statement precise...
Let $C$ be a first-order axiomatizable class of structures (in a countable language) which is "categorical in $2^{\aleph_0}$" -- i.e. any two structures in $C$ of size continuum are isomorphic. Then $C$ has the S-B property with respect to elementary embeddings. (This generalizes the cases of vector spaces and algebraically closed fields.)
Addendum: A completely different way that a category $C$ might be Schroeder-Bernstein is if every object is "surjunctive" (i.e. any injective self-morphism of an object is necessarily surjective). This covers Justin's example of the category of well-orderings.
Let me point out a curious (non-categorical) twist to an elementary observation. The elementary observation is that the Cantor-Schroeder-Bernstein property fails for linear orderings, even if we require the injections to map onto an initial segment. That is, you can have two non-isomorphic linear orders, each isomorphic to an initial segment of the other. One example is the same as a familiar example for the topological case, the closed interval [0,1] and the half-open interval [0,1) of real numbers. And of course, the situation is the same for final segments. The curious twist is that, if a linear order A is isomorphic to an initial segment of another linear order B, while B is isomorphic to a final segment of A, then A and B are isomorphic.
We discussed this at the Secret Blogging Seminar.