Since $\left(\frac{16}{2\sqrt{3}}\right)^2 = \frac{64}{3} = 21+\frac{1}{3}$ lies between $4.5^2 = 20.25$ and $5^2$ and the altitude of the triangle is $\frac{2\sqrt{3}\cdot\sqrt{3}}{2} = 3$, we can cover a square with sidelength $16$ with $2 \cdot 5 \cdot \lfloor\frac{16}{3}\rfloor = 60$ equilateral triangles. Since $\lfloor\frac{1000}{60}\rfloor = 16$, by the pigeon hole principle there is an equilateral triangle that covers at least $17$ points.