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Jul 24 at 12:10 comment added Andrew @Sturrum That seems like an overly broad interpretation of the OP's intent to me. But regardless, I actually don't think the issue you bring up is that complicated. Classical mechanics succeeds whenever it agrees with experiment. It fails to describe Hydrogen because it predicts the atom would be unstable, which does not agree with experimental evidence about Hydrogen. Broadly speaking, classical mechanics tends to work in processes involving large numbers of particles at high temperatures. But ultimately whether classical mechanics works is an empirical question.
Jul 24 at 8:04 vote accept Polaris5744
Jul 24 at 8:03 comment added Sturrum Despite the fact that this was accepted, I don't think it actually answers the question. It gives a particular case where classical mechanics fails, but it does not describe what criteria a system needs to meet for classical theory to work or fail, which much more complicated.
Jul 23 at 14:52 vote accept Polaris5744
Jul 24 at 8:04
Jul 23 at 12:47 history answered Andrew CC BY-SA 4.0