In example $1.14$ on page $51$ (of the book and $64$ of this link), shouldn't the string $01000$ get rejected? However it seems that the first three digits of the string would force it to an accept state. Would the final digit push it out of an accept state?
Indeed, $01000$ will be rejected. The first four digits $0100$ will get the NFA into states $q_1$ and $q_4$. But, as you correctly noticed, the final digit $0$ will kill the copy at state $q_4$. The only state left will be $q_1$, and it is not an accepting state, so the string is rejected.