Here is an example for Torsten's Addendum: a line bundle with the required properties which is not semi-ample (no multiple is globally generated):
Take an $\pi:X\to \mathbb P^n$ as in Torsten's example, that is, with $\pi_*\mathcal O_X\simeq \mathcal O_{\mathbb P^n}$. Also assume that there is an exceptional divisor $E$, so let's just say it is a birational morphism contracting the divisor $E$ to a point. Now take $\mathcal L=\pi^*\mathcal O_{\mathbb P^n}(1)$ and consider the short exact sequence:
$$ 0\to \mathcal L^{\otimes n} \to (\mathcal L(E))^{\otimes n} \to \mathcal O_{nE}(nE) \to 0 $$ The sheaf on the right has no global sections, so the other two have the same $H^0$. Therefore $$ H^0(X,(\mathcal L(E))^{\otimes n}) = H^0(\mathbb P^n, \mathcal O_{\mathbb P^n}(n)) $$
Finally $(\mathcal L(E))^{\otimes n}$ cannot be globally generated, because $H^0(X,\mathcal L^{\otimes n})=H^0(X,(\mathcal L(E))^{\otimes n})$, so ${\rm supp}\\,E$ is always in the base locus (or in other words, if it had a section whose zero section did not contain ${\rm supp}\\,E$, then its restriction to $nE$ would give a nonzero section of $\mathcal O_{nE}(nE)$).