7. The minimum possible value of $n$ is 911.
(1) First, prove: 911 satisfies the condition.
If each student answers at least three questions correctly, since the number of different ways a student can answer three questions correctly is $\mathrm{C}_{15}^{3}=455$, then if there are 911 students participating, by the pigeonhole principle, at least 3 students must have answered the same three questions correctly.

If a student answers no more than 2 questions correctly, then among the remaining students, the number of students who answer no more than three questions correctly cannot exceed 10 (otherwise, the total score of these students and the first student would be less than 36 points). For the remaining $911-11=900$ students, each student answers at least 4 questions correctly.

Since $\mathrm{C}_{4}^{3} \times 900 > 455 \times 2$, there must be at least 3 students who answered the same three questions correctly.
(2) If there are 910 students participating, these students can be divided into 455 groups, with each group consisting of two students who answered the same three questions correctly. In this case, the condition of the problem is not satisfied.
In conclusion, $n_{\min }=911$.