If not, when can you not comply with it?
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No, because this is a sufficient condition (for regularly sampled signals), and not a necessary one. This condition restricts the space of all possible continuous signals to a subspace of discrete sequences that contain the same information.
Suppose that you can constrain the signal space, eg limited band-width, positivity, parametric models, sparsity, etc. Then many theoretical works ensure perfect reconstruction (in theory) under Nyquist-like conditions. Literature abounds, under the names of compressive or compressed sensing, sparse sampling, finite-rate of innovation, etc. Some are "exactly exact", some are exact within a high probability, under quantization bounds, etc.
The topic is very wide and active, I cannot provide more details unless the question is more precise. A couple of papers however:
answered Dec 12 '19 at 8:25
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