Source Coding Theorem: what happen when we go below N*H(x) bits?
It's not clear exactly what your question is. What the textbook is talking about is the strong converse to Shannon's source coding theorem, but rate distortion theory is another part of information theory relevant to this.

The strong converse says that if you try to compress information to fewer bits than the entropy, the probability that all of the information is restored correctly goes to zero as $N$ goes to $\infty$ (the theorem also gives you a bound on this probability).

Rate distortion theory answers the question: I have a signal that I want to compress to a length that is shorter than its entropy. How small an amount of distortion can I introduce into the signal to ensure that I can recover it with high probability? You can use this theory with an arbitrary formula for distortion (and there is a nice formula if the distortion is given by a single-symbol equation). 

And of course, if you compress the source to $K$ bits, than you can choose $K/H(X)$ symbols of the original to keep and discard all the others.