Let $X$ be a complex K3 surface and $C$ a smooth curve on $X$ and $A$ a basepoint free line bundle on $C$.

Aprodu's paper - Lazarsfeld Mukai bundles and applications says this. We cannot lift the linear system $|A|$ to $X$ if $Pic\,X$ is generated by $C$ or if $X$ contains no elliptic curves, for most $|A|$.

I do not understand this statement. I know that in general $A$ need not be a restriction of a line bundle from $X$. But he seems to be saying more. Any clarification of the above statement will be helpful.