Example 3 There are $n$ students in a class, who are to participate in a table tennis doubles match. The class held a warm-up match first. These $n$ students paired up with each other, and each pair participated in one warm-up match exactly. Please write down all possible values of $n$, and provide a match arrangement for one of the cases.
(2012, Joint Autonomous Admission Examination of Tsinghua University and Other Universities)
Assume that $k$ warm-up matches have been held.
Since two pairs of students appeared in one match, $\mathrm{C}_{n}^{2}=2 k \Rightarrow 4 k=n(n-1)$.
Thus, $n=4 l$ or $n=4 l+1\left(l \in \mathbf{N}_{+}\right)$.
Next, we prove that when $n=4 l$ or $n=4 l+1\left(l \in \mathbf{N}_{+}\right)$, a match schedule that meets the requirements can be constructed.
We apply mathematical induction on $l$.
(1) When $l=1$, let the 5 students be $a, b, c, d, e$. A possible match schedule is
$a b \leftrightarrow c d, a c \leftrightarrow b e, b c \leftrightarrow d e, a e \leftrightarrow b d, a d \leftrightarrow c e$.
(2) Assume that the conclusion holds when $l=m$.
When $l=m+1$, suppose the $4 m+5$ players are $a, b, c, d, e, f_{1}^{1}, f_{1}^{2}, f_{2}^{1}, f_{2}^{2}, \cdots, f_{2 m}^{1}, f_{2 m}^{2}$.
By the induction hypothesis, we can arrange the matches between
$e, f_{1}^{1}, f_{1}^{2}, f_{2}^{1}, f_{2}^{2}, \cdots, f_{2 m}^{1}, f_{2 m}^{2}$

such that every two students are partners and participate in exactly one warm-up match.
At this point, there are still matches to be arranged between $a, b, c, d$ and
$e, f_{1}^{1}, f_{1}^{2}, f_{2}^{1}, f_{2}^{2}, \cdots, f_{2 m}^{1}, f_{2 m}^{2}$.

The match schedule between $a, b, c, d, e$ is known to exist from (1).

Then, the match schedule between $a, b, c, d$ and $f_{1}^{1}, f_{1}^{2}, f_{2}^{1}, f_{2}^{2}, \cdots, f_{2 m}^{1}, f_{2 m}^{2}$ can be arranged as
$a f_{i}^{1} \leftrightarrow b f_{i}^{2}, a f_{i}^{2} \leftrightarrow b f_{i}^{2}, c f_{i}^{1} \leftrightarrow d f_{i}^{2}, c f_{i}^{2} \leftrightarrow d f_{i}^{1}$.
Thus, when $l=m+1$, the condition is also satisfied.
When $n=4 l$, it can be similarly proven.
In summary, all possible values of $n$ are
$4 l$ or $4 l+1\left(l \in \mathbf{N}_{+}\right)$.
[Note] The prototype of this problem is the 19th Russian Mathematical Olympiad problem from 1993. The original problem is:

Suppose there are $n$ tennis players who play doubles matches with the rule that each player plays exactly one match against every other player as an opponent. For which natural numbers $n$ can such matches be arranged?
Hint: $n$ satisfies $n \equiv 1(\bmod 8)$.