Let $X$ be a smooth complex projective variety of dimension $n$, and $E\in H^{n-k}(X,\mathbb{Q})$. We have the operation of intersecting with a hyperplane class $H$, i.e. $- \cup H \colon H^i(X, \mathbb{Q}) \to H^{i+2}(X, \mathbb{Q})$. I have the following definitions of primitive/non-primitive cohomology:
$H^{n-k}_{\mathrm{nprim}}(X, \mathbb{Q}):=\mathrm{im}(-\cup H \colon H^{n-k+2}(X, \mathbb{Q}) \to H^{n-k}(X, \mathbb{Q}) )$.
$H^{n-k}_{\mathrm{prim}}(X, \mathbb{Q}) := \ker(-\cup H^{k+1} \colon H^{n-k}(X, \mathbb{Q}) \to H^{n+k+2}(X, \mathbb{Q}) )$.
If $E\cup H=0$, then $E\cup H^{k+1}=0$, so $E\in H^{n-k}_{\mathrm{prim}}(X,\mathbb{Q})$. If $E\neq 0$, do we know what $k$ can be (without knowing anything about the Hodge diamond of $X$)?