Example 6 Let real numbers $a, b, c, d, e$ satisfy
$$a+b+c+d+e=8, a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16,$$

Find the maximum value of $e$.
Solve: Rewrite the conditions as
$$8-e=a+b+c+d, 16-e^{2}=a^{2}+b^{2}+c^{2}+d^{2},$$

From this, we get an inequality involving $e$. By the Cauchy-Schwarz inequality, we have
$$a+b+c+d \leqslant(1+1+1+1)^{\frac{1}{2}}\left(a^{2}+b^{2}+c^{2}+d^{2}\right)^{\frac{1}{2}} .$$

Substitute the conditions and square both sides, we get
$$\begin{array}{c}
(8-e)^{2} \leqslant 4\left(16-e^{2}\right) \\
64-16 e+e^{2} \leqslant 64-4 e^{2} \\
5 e^{2}-16 e \leqslant 0, e(5-16 e) \leqslant 0
\end{array}$$

From this, we get $0 \leqslant e \leqslant \frac{16}{5}$, and the maximum value $\frac{16}{5}$ is achieved when $a=b=c=d=\frac{6}{5}$.