I generally hear it assumed that Bell's inequality implies violation of counterfactual definiteness, because locality is considered sacrosanct. I understand of course that measurable violations of locality are logically inconsistent. But what is so bad about "hidden" violations of locality? What are the reasons nonlocal hidden variable theories are frowned upon? Is it just because the ontologies currently on the table (such as de Broglie–Bohm theory) are considered kind of ugly?
Locality is emphasized, because without locality, it used to be thought that physics would be too arbitrary. DeBroglie Bohm picks out a preferred basis, namely the x-basis, to make particles run around in, and you could do this in a bosonic field basis, in a rotated field basis, or a billion ways. So it isn't correct to call it a theory, it is a procedure for producing hidden variables given a quantum model and a basis.
String theory, and the holographic principle more generally, moots the locality debate. Gravity is nonlocal, the spacetime interior to a black hole is reconstructed from surface states, and there is no more argument against local hidden variables.
I believe that t'Hooft revived hidden variable models for exactly this reason in the 1990s--- the old arguments for locality disappear in light of holography.
Luboš as always gives a good account. There are many alternate accounts, however, some of which make some sense. Your question is asked in a way that suggests to me a specific type of answer.
Einstein locality of the dynamics is very well supported by experiment. If by locality you mean Einstein locality, then there are no "measurable violations of locality". On the other hand, there is no locality of initial conditions, by definition; consider, for example, a classical field that is zero everywhere in Minkowski space, or, equally nonlocally, the vacuum state of quantum field theory, which is by definition the same wherever and whenever you measure it. This kind of nondynamical nonlocality is the basis of one of the many ways of evading the derivation of Bell inequalities for random fields, which is usually pejoratively dismissed as the "conspiracy" loophole, but which, nonetheless, is there. [BTW, the conspiracy requires only a dynamically local deterministic evolution of probability distributions, not a deterministic evolution of trajectories.] Now, if you like, this is "hidden" nonlocality because it's "nondynamical", but I doubt almost anyone thinks there's anything wrong with it, insofar as we work with initial conditions all the time.
The distinction I make above between locality as a property of a dynamics and locality as a property of an initial condition is only one of many fine distinctions that have been made in the literature. Be careful how you use the word "locality".
Counterfactual definiteness has definitely been much made of in the literature on Bell inequalities for the particle case. The same idea (or perhaps it's just similar) can be put, less Philosophically, in terms of noncontextuality, the idea that one shouldn't have to say what experimental apparatus was used to measure a property. In these terms, the contextuality that is required to model Bell violating experiments classically is nonlocal in the sense that the whole measurement apparatus interacts both with the purported system that is measured and with the whole preparation apparatus, even if the system that is measured is purported to be two particles at the opposite ends of a light-years long fiber optic.
As a postscript to the above, which might or might not be a useful answer, according to your taste, the only way I have found to make this unproblematic is to take the "purported system" to be a (random) field in a coarse-grained equilibrium state (coarse-grained in the sense that the statistics of measurement events are invariant under time-like translations --in the sense, say, that they must be repeatable to get into a journal--, even though the events themselves are clearly not manifestations of a fine-grained equilibrium). Since the field is everywhere in the apparatus, and it's a commonplace that an equilibrium state is a nonlocal accommodation to whatever boundary conditions have been put in place by the experimenter, a nondynamical nonlocality is to be expected. Note too that the larger the experimental apparatus is, the longer we have to wait before we will record measurement events at the measurement apparatuses and the longer it will take to verify that the measurement events at the two ends in fact violate Bell inequalities (and the harder it will be to ensure that they do, despite the environment). Although it goes far into details that I won't expound here, particles in this view are modulations of the Vacuum Expectation Values, a generalization to the random field context of modulations of a classical field.
A large part of this approach is to apply ideas from quantum field theory as if they are signal processing mathematics. The violation of Bell inequalities can be derived for random fields only with assumptions that are not natural for a random field, whereas the assumptions required to derive the violation of Bell inequalities for classical particle models are generally deemed rather natural by most Physicists. It is, however, somewhat strange to ask classical physics not to use the resources of a random field.
The reason why locality has to hold in physics isn't a religious belief that could be associated with the word "sacrosanct". Instead, the reason is a fully technical set of insights known as the special theory of relativity that was found by a physicist named Albert Einstein in 1905. The physicist is rather famous but the main content of his theories is not known and the original question is an example of that.
According to relativity, influences faster than light – e.g. immediate action at a distance – are strictly forbidden because from a different inertial frame, they would become influences that affect the past and such influences propagating to the past would lead to logical contradictions. This statement is completely universal, whether or not we talk about microscopic, macroscopic, classical, or quantum phenomena.
The reasons why the EPR-like entanglement experiments behave the way they behave have nothing whatsoever to do with nonlocality. The correlation between the entangled EPR subsystems isn't a result of any nonlocal influences during the measurements; instead, as quantum mechanics (which is experimentally confirmed beyond any doubt) makes totally indisputable, the correlations follow from the contact of the subsystems sometime in the past when their entangled state was created.
Not only nonlocality has nothing to do with the explanation of the results of EPR experiments. Also, a class of quantum mechanical theories known as quantum field theory (QFT) respect locality totally accurately, in all the evolution, because they respect the Lorentz symmetry and locality follows from the Lorentz symmetry (special relativity), as explained at the beginning.
Bell's theorem in combination with the measured correlations outside the Bell's interval falsify "local realist" theories. However, it may be seen either by the relativistic argument above or by more detailed reasoning based on additional experiments that it's realism, and not locality, which is the wrong assumption making it impossible for local realist theories to agree with observations. Both locality and non-realism are needed for a viable theory; locality is needed since 1905 because of relativity while non-realism has been needed since 1925 when the quantum revolution took place. In particular, quantum field theory (and string theory) which is the state-of-the-art framework describing (almost) all observations is an exactly local but quantum (i.e. non-realist) theory.
An example of an explicit experiment that falsifies a huge class of hypothetical nonlocal realist theories – in a similar way in which Bell's theorem falsifies local realist theories – is described e.g. in this paper from 2007:
Some more comments why most things that people say about "nonlocal theories" are indefensible at the level of genuine physics, despite the widespread confused statements about "nonlocality" of our world in the popular science literature, may be found e.g. at