【Analysis】According to the problem, a "good number" can be expressed as the sum of two non-zero natural numbers $m, m+1$, and also as the sum of three non-zero natural numbers $n, n+1, n+2$. Therefore, we can set up the equation $2 m+1=3 n+3$, and derive the range of $n$, thereby solving the problem.

【Solution】Solution: A "good number" can be expressed as the sum of two non-zero natural numbers $m, m+1$, and also as the sum of three non-zero natural numbers $n, n+1, n+2$.

So $2 m+1=3 n+3$, which means $m=\frac{3}{2} n+1$, so $2 \mid n$.
Since $3 n+3 \leqslant 2011$, we have $n \leqslant 669 \frac{1}{3}$,
Since $2 \mid n$, the maximum value of $n$ is 668,
At this point, $3 n+3=2007$.
The largest "good number" not exceeding 2011 is 2007.
Therefore, the answer is: 2007.
5. (10 points) A natural number that can be expressed as the sum of two consecutive non-zero natural numbers, and also as the sum of three consecutive non-zero natural numbers, is called a "good number". The largest "good number" not exceeding 2011 is . $\qquad$