sampling a signal below nyquist rate
No, recovering an "arbitrary" signal from its samples taken below its Nyquist rate (assuming it is bandlimited) is not possible in general. 

Compressive sensing is based on the assumption that if you know a-priori that a signal will be sparse in some of its support domains, then those bands will not be carrying information about the signal and can be discarted in principle, therefore, reducing the pre-computed Nyquist sampling rate, which was computed based solely on the bandlimitedness (Shannon-Nyquist sampling theorem) of the signal.

The trick here is on the "a-priori" knowledge which also contains information, and this must be taken into account when computing the total information present in a signal. This a-priori knowledge can be the result of imposed structures on the signal generation mechanism, or well known models of the signal source.

If you can impose no constraints on the signal other than its bandlimit, you have to sample it above its Nyquist rate, so that exact reconstruction (using unimplementable ideal lowpass reconstruction filters) is possible at least in principle. 

The topic of sampling is bound by information theory. There is no way to walk around it. Why would you ever sample a DC signal, or a single sine wave, or a periodic signal? It is based what is known and what is not.