Suppose $n$ quadric hypersurfaces cut out $2^n$ distinct points $p_1,\ldots,p_{2^n}$ in $\mathbb{P}^n$. What is the maximal number of points $p_i$ a quadric can contain without containing all of them?

For n=2, there is of course a quadric going through any three points and avoiding the last point. In $\mathbb{P}^3$, it is easy to exhibit a quadric going through 6 points (and avoiding the last two). This is however not possible with 7 points (as can be seen using a projection to $\mathbb{P}^2$).