Consider $H_1,H_2,H_3\subset\mathbb{P}^{2m+1}$ three general linear subspaces of projective dimension $m$.

Then there exists a quadric hypersurface $Q^{2m}\subset\mathbb{P}^{2m+1}$ containing $H_1,H_2,H_3$. This is just a dimension count.

*Does there exist a smooth quadric hypersurface $Q^{2m}\subset\mathbb{P}^{2m+1}$ containing $H_1,H_2,H_3$?*

The answer is positive when $m = 1$, since if $Q^2$ is a cone then the three lines must intersect, and if $Q^2$ is the union of two planes or a double plane then the three lines must intersect as well.