16. A competition consists of $n$ true/false questions. Statistics on the answers of 8 candidates show that: for any two questions, exactly two candidates answer “ $T, T$ ”; exactly two candidates answer “ $F, F$ ”; exactly two candidates answer “ $T, F$ ”; exactly two candidates answer “ $F, T$ ”. Find the maximum value of $n$.
Let $T$ be $1$ and $F$ be $0$, then we can get an 8-row $n$-column number table.
Obviously, swapping the 0s and 1s in the same column does not change the properties of this table. Without loss of generality, assume the first row is all 0s. Let the $i$-th row have $a_{i}$ zeros $(i=1,2, \cdots, 8)$, then $a_{1}=n, \sum_{i=2}^{8} a_{i}=4 n-n=3 n$.
Next, we calculate the number of 00 pairs in each row.
According to the problem, for any two questions, exactly two candidates answer “$F, F$”, which means the number of 00 pairs is $2 C_{n}^{2}$; on the other hand, the number of 00 pairs is $\sum_{i=1}^{8} C_{a_{i}}^{2}$, thus $\sum_{i=1}^{8} C_{a_{i}}^{2}=2 C_{n}^{2} \Rightarrow \sum_{i=2}^{8} C_{a_{i}}^{2}=C_{n}^{2}$. Therefore, $\sum_{i=2}^{8} \frac{a_{i}\left(a_{i}-1\right)}{2}=\frac{n(n-1)}{2} \Rightarrow \sum_{i=2}^{8} a_{i}^{2}-\sum_{i=2}^{8} a_{i}=n^{2}-n$ $\Rightarrow n^{2}+2 n=\sum_{i=2}^{8} a_{i}^{2} \geqslant \frac{1}{7}\left(\sum_{i=2}^{8} a_{i}\right)^{2}=\frac{9 n^{2}}{7} \Rightarrow n \leqslant 7$.
When $n=7$, construct the following table:
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\hline 0 & 1 & 1 & 1 & 1 & 0 & 0 \\
\hline 0 & 1 & 1 & 0 & 0 & 1 & 1 \\
\hline 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
\hline 1 & 0 & 1 & 0 & 1 & 0 & 1 \\
\hline 1 & 0 & 1 & 1 & 0 & 1 & 0 \\
\hline 1 & 1 & 0 & 0 & 1 & 1 & 0 \\
\hline 1 & 1 & 0 & 1 & 0 & 0 & 1 \\
\hline
\end{tabular}

In summary, the maximum value of $n$ is 7.