I read the article by Mermin which @joshphysics refers to in his comment, and I have to say it is a very good article. It gets to the heart of Bell's paradox in a very clear way. But it contains two fallacies which I think ought ought to be pointed out.
The first is the idea that a Stern-Gerlach apparatus can look a single atom and measure it as either spin-up or spin-down. This is not how magnets actually work. Any magnetic field that is inhomogenous in the x direction is just as inhomogenous in the y direction. So it cannot split a beam into x-up vs x-down components without also splitting into y-up and y-down components. Yes, Stern and Gerlach did something like this, but to a fan-shaped beam, not a pencil beam. (I am assuming the standard Stern-Gerlach experiment with propagation in the z-direction.) Mermin talks about an abstract machine with three switch settings, but at the end he admits he is talking about Stern-Gerlach. Doesn't work that way. I explain this in more detail in my blogpost Quantisation of Spin Revisited.
A bigger problem is Mermin's "machine" which lets you push a button and then out come two entangled particles. I don't think such a machine exists. Certainly, the experiments which people talk about with spontaneous down-conversion and coincidence counters are much more complicated than this. I know that people who know better than me will say that it comes to exactly the same thing, but I'm not sure about that. I wrote an article about this once called "There Are No Pea-Shooters for Photons". I don't think there is a pea-shooter for photons, and I'm pretty damn sure that Mermin's pea-shooter for entangled spin pairs does not exist.
There is one more problem with Mermin's analysis, which is not exactly a problem because it's a very clever thing that we owe to Bell in terms of logical clarity. But it obscures the real physics. I'm talking about the notion that the real contradiction occurs when we skew the detectors. That there's nothing wrong with Case A, where coincidences are detected with a probability of 100%.
In fact, that's already a hell of a problem if we have both detectors parallel and we get 100% correlation. Yes, I know you think that just means that the particles were created in a correlated state...so what's the problem? The problem is that yes, if HAD a pea-shooter (which you don't) that created projectiles in pairs, spin-up and spin-down, and if you HAD a Stern-Gerlach machine to positively detect those spins (which you don't)...if you had those things, yes, you should expect 100% correlation at the detectors. But any physically conceivable "pea-shooter" cannot and will not reliably produce projectile pairs correlated in the x direction only. The pea-shooters you can realistically imagine will produce correlated pairs in all possible orientations. And the ideal Stern-Gerlach machines (the ones that don't exist) will not give you 100% correlation on those types of sources. Because if you produce a y-correlated pair and put them into an x-aligned "ideal" Stern-Gerlach detector, you will get a 50% conincidence rate.
Any realistic source will produce randomly-correlated projectile pairs, so the "ideal" detector pairs cannot be 100% correlated. Bell's paradox provides a brilliant answer the very esoteric philosophical question of "what if you DID have such a magical source", but from a practical point of view, the horse is already out of the barn if you can show 100% correlation from an ordinary source.
There are people who know more than me about these things, but I do not believe they have answers to my arguments.