Let $X$ be a non-singular projective variety, and $D$ a divisor on $X$.

Saying that $D$ has positive (meaning non-zero) Iitaka dimension is equivalent to the function $n \mapsto h^0(\cal{O}(D))$ being strictly increasing for sufficiently large $n$?

Does every effective divisor have positive Iitaka dimension? If not, what are counterexamples?