Assume that ${\rm char}=0$. If $X$ is Fano and $\mathcal L$ is an ample invertible sheaf contained, but not equal to $\omega_X^{-1}$, then $H^i(X,\mathcal L^{-1})=0$ for all $i$. This is Kodaira vanishing for $i\neq \dim X$ and for $n=\dim X$, $H^n(X,\mathcal L^{-1})$ is dual to $H^0(X, \mathcal L\otimes \omega_X)$ which is $0$ by the choice of $\mathcal L$. 

I think both of your examples are covered by this.

In general, if $\mathcal L$ is contained in $\omega_X$, then $H^n(X,\mathcal L)\neq 0$, so for a K3 (or more generally if $K_X=0$) negative ample invertible sheaves will not work.