Let A be a non-isotrivial family of abelian varieties over a smooth irreducible curve S. Suppose the generic fiber of A is simple and call R its endomorphism ring. We consider an irreducible curve C in the n-fold fibered power of A and suppose that everything is defined over a number field k. Then C defines n points P1,...Pn points on A(k(C)). Then, there are at most finitely many points c on the curve such that the specialized P1(c),...Pn(c) are dependent over R, unless they were already identically dependent. This, combined with earlier works of the authors and of Habegger and Pila, gives a general unlikely intersections statement for (not necessarily simple) families of abelian varieties. The proof of these theorems uses a method introduced by Pila and Zannier and combines results coming from o-minimality with some Diophantine ingredients. These results have applications to the study of the solvability of some Diophantine equations in polynomials.