jmc

Since $15\mid 15n$, and $15$ is square-free, we must have $15^2\mid 15n$, so $15\mid n$. Say $n=15a$. Then $15^{2}a=15n$ is a square, and conversely, if $a$ is a square, then $15^{2}a$ is a square. Thus we are counting the number of positive squares $a$ such that $15a\le 1000$ or $a\le \frac{200}{3}\approx 66.6$. The largest such square is $64=8^2$, so the possible values of $a$ are $b^2$ for $b=1,2,3,4,5,6,7,8$, giving $\boxed{8}$ possible values of $a$ (and hence 8 possible values for $n$).