Let A be an abelian variety over a number field K, with A(K) Zariski-dense in A. In this talk I will show that for every irreducible ramified cover π : X → A the set A(K) \ π (X(K)) of K-rational points of A that do not lift to X(K) is still Zariski-dense in A, and that in fact it even contains a finite-index coset of A(K). This result is motivated by Lang's conjecture on the distribution of rational points on varieties of general type and confirms a conjecture of Corvaja and Zannier concerning the "weak Hilbert property" in the special case of abelian varieties.