To prove that $n \leq 18$, we will use a combinatorial argument based on the given constraints.

1. Counting Pairs in Clubs:
Each club consists of 4 students. The number of ways to choose 2 students out of 4 is given by the binomial coefficient:
$$\binom{4}{2} = 6$$
Therefore, each club contributes 6 pairs of students.

2. Total Pairs in All Clubs:
If there are $n$ clubs, the total number of (not necessarily distinct) pairs of students in all clubs is:
$$6n$$

3. Counting Distinct Pairs:
There are 9 students in total. The number of distinct pairs of students is:
$$\binom{9}{2} = 36$$

4. Average Number of Clubs per Pair:
If each pair of students can be in at most 2 clubs together, the maximum number of pairs that can be formed without violating the given condition is:
$$2 \times 36 = 72$$
This is because each of the 36 pairs can appear in at most 2 clubs.

5. **Inequality for $n$**:
For $n$ clubs, the total number of pairs (6n) must be less than or equal to 72:
$$6n \leq 72$$
Solving for $n$:
$$n \leq \frac{72}{6} = 12$$

6. Conclusion:
Therefore, the maximum number of clubs $n$ that can be formed under the given constraints is 12. This shows that the bound $n \leq 18$ is indeed correct, but we have shown a tighter bound of $n \leq 12$.

$\blacksquare$

The final answer is $\boxed{ n \leq 12 }$