In their paper "Monge-Ampère equations in big cohomology classes", Boucksom, Eyssidieux, Guedj and Zeriahi give an example (Ex 5.4 page 46 here : http://arxiv.org/abs/0812.3674)
of a nef and big line bundle over a smooth projective 3-fold which is not semi-ample.
More precisely, every positive current in its cohomology class has poles along some subvariety.

Furthermore, it is well-known (Lazarsfeld, PAG e.g) that a nef and big line bundle has a finitely generated sections ring iff it is semi-ample.

In one word, their construction consists in using the famous example of Serre (and studied by Demailly-Peternell-Schneider) of a flat rank 2 vector bundle $E$ on some elliptic curve $C$, and considering on $V:=\mathbb P(E\oplus A)$ (for $A$ ample on $C$) the tautological line bundle $\mathcal O_{\mathbb P(V)}(1)$.