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 projective 3-fold which is not semi-ample. Even better, their is only one positive current in its cohomology class, and it has pole along some subvariety.
Furthermore, it is well-known (Lazarsfeld, PAG e.g) that a nef and big line bundle has a finite generated sections ring iff it is semi-ample.