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 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.