  A multipartite quantum state violates a Bell inequality asymptotically if,
after jointly processing by general local operations an arbitrarily large
number of copies of it, the result violates the inequality. In the bipartite
case we show that asymptotic violation of the CHSH-inequality is equivalent to
distillability. Hence, bound entangled states do not violate it. In the
multipartite case we consider the complete set of full-correlation Bell
inequalities with two dichotomic observables per site, also called
WWZB-inequalities. We show that asymptotic violation of any of these
inequalities by a multipartite state implies that pure-state entanglement can
be distilled from it, although the corresponding distillation protocol may
require that some of the parties join into several groups. We also obtain the
extreme points of the set of distributions generated by measuring $N$ quantum
systems with two dichotomic observables per site. It is shown that when
considering the violation of any Bell inequality after preprocessing, either
deterministic LOCC or stochastic local operations (without communication) is
enough.
