Example 3 The condition for three line segments to form a triangle is: the sum of the lengths of any two line segments is greater than the length of the third line segment. There is a wire of length $144 \mathrm{~cm}$, which is to be cut into $n$ $(n>2)$ small segments, with each segment being no less than $1 \mathrm{~cm}$ in length. If any three of these segments cannot form a triangle, then the maximum value of $n$ is $\qquad$
Since the necessary and sufficient condition for forming a triangle is that the sum of any two sides is greater than the third side, the condition for not forming a triangle is that the sum of any two sides is less than or equal to the largest side. The shortest piece of wire is 1, so we can place 2 ones, and the third segment is 2 (to maximize $n$, the remaining wire should be as long as possible, so each segment is always the sum of the previous two segments), in sequence:
$$
1,1,2,3,5,8,13,21,34,55 \text {, }
$$

The sum of the above numbers is 143, which is 1 less than 144. Therefore, the last segment can be 56, at which point $n$ reaches its maximum of 10.