Solution: 33.
By the convexity of the sine function, we have
when $x \in\left(0, \frac{\pi}{6}\right)$, $\frac{3}{\pi} x < \sin x < x$,
$$
\frac{3}{\pi} \times \frac{\pi}{12}=\frac{1}{4},
$$
$$
\sin \frac{\pi}{10} < \frac{3}{\pi} \times \frac{\pi}{9}=\frac{1}{3} \text {. }
$$

Therefore, $\sin \frac{\pi}{13}<\frac{1}{4}<\sin \frac{\pi}{12}<\sin \frac{\pi}{11}<\sin \frac{\pi}{10}<\frac{1}{3}<\sin \frac{\pi}{9}$.
Thus, the positive integer values of $n$ that satisfy $\frac{1}{4}<\sin \frac{\pi}{n}<\frac{1}{3}$ are $10, 11, 12$, and their sum is 33.
7. The sum of all positive integers $n$ that satisfy $\frac{1}{4}<\sin \frac{\pi}{n}<\frac{1}{3}$ is