Is it sort of like the n-body problem of classic chaotic behavior?
Regarding the first question, I am only repeating pop-science here.
I've read (In something by Roger Penrose, presumably The Emperors New Mind, but maybe in The Road to Reality) that the Maxwell Equations are uncomputable only if the initial conditions are functions that don't have Fourier transforms.
So really it is the field dynamics that are non-computable. I might have been talking nonsense when I the fields themselves were non-computable. But then again, the kinds of functions that don't have a Fourier transforms are also very rough, jaggedy things that probably can't be approximated by any algorithm.
And speaking of rough jaggedy functions, that's what you get if you evolve a chatoic system to infinity. So maybe there is some connection after all, but it is well beyond my understanding.
So that kind of computability might be the mathematical algebraic computability (as in does this monstrosity have a closed form?), which can always be approximated (of course with fast accumulating errors).
For example, classical field equations apparently can behave in non-computable ways, so long as their initial conditions are sufficiently wierd. I'm not sure if QM as we undestand it can handle such fields: but then "as we understand it" is a bit too limiting.
So why haven't we observed these wierd non-computable fields? My guess is that it's because they don't exist. But it might be that we just haven't recognized them, perhaps some of those puzzling things in physics -- like QM perhaps itself is caused by naughty non-computable stuff going on where we haven't yet recognized it.