Example 1 Given $x_{1}<x_{2}<\cdots<x_{7}$, and all are positive integers, $x_{1}+x_{2}+\cdots+x_{7}=158$. Find the maximum value of $x_{1}+x_{2}+x_{3}$.
Estimate the maximum value of $x_{1}+x_{2}+x_{3}$ to be 60.
First, prove: $x_{1}+x_{2}+x_{3} \leqslant 60$.
Assume $x_{1}+x_{2}+x_{3} \geqslant 61$. Then
$$
\begin{array}{l}
\left(x_{3}-2\right)+\left(x_{3}-1\right)+x_{3} \geqslant x_{1}+x_{2}+x_{3} \geqslant 61 \\
\Rightarrow x_{3} \geqslant 22 \\
\Rightarrow x_{4}+x_{5}+x_{6}+x_{7} \\
\quad \geqslant x_{3}+1+x_{3}+2+x_{3}+3+x_{3}+4 \\
\quad=4 x_{3}+10 \geqslant 4 \times 22+10=98 \\
\Rightarrow x_{1}+x_{2}+\cdots+x_{7} \\
\quad=\left(x_{1}+x_{2}+x_{3}\right)+\left(x_{4}+x_{5}+x_{6}+x_{7}\right) \\
\quad \geqslant 61+98=159,
\end{array}
$$

which contradicts the given equation.
Therefore, $x_{1}+x_{2}+x_{3} \leqslant 60$, i.e., the maximum value of $x_{1}+x_{2}+x_{3}$ is 60.
Next, construct an example to show that 60 can be achieved.
Take $x_{1}=19, x_{2}=20, x_{3}=21, x_{4}=22$,
$$
x_{5}=23, x_{6}=24, x_{7}=29 \text {. }
$$

Then $x_{1}+x_{2}+x_{3}=60$.