Example 7 Find the smallest positive integer $n$, such that the indeterminate equation
$$x_{1}^{4}+x_{2}^{4}+\cdots+x_{n}^{4}=1599$$

has integer solutions $\left(x_{1}, x_{2}, \cdots, x_{n}\right)$.
Note that for any $x \in \mathbf{Z}$, if $x$ is even, then $x^{4} \equiv 0(\bmod 16)$; if $x$ is odd, then $x^{2} \equiv 1(\bmod 8)$, and in this case, $x^{4} \equiv 1(\bmod 16)$.

The above discussion shows that $x_{i}^{4} \equiv 0$ or $1(\bmod 16)$, thus the remainder of $x_{1}^{4}+x_{2}^{4}+\cdots+x_{n}^{4}$ modulo 16 lies between 0 and $n$. Therefore, when $n \leqslant 14$, equation (9) cannot hold (since $1599 \equiv 15(\bmod 16)$).

Notice that $5^{4}+12 \times 3^{4}+1^{4}+1^{4}=1599$, so when $n=15$, equation (9) has an integer solution $(5, \underbrace{3, \cdots, 3}_{12 \uparrow}, 1,1)$.

In summary, the smallest positive integer $n=15$.