I apologise for the basic question; I am reading Huybrecht's Lecture Notes on K3 surfaces, and on p.257 it is mentioned an example of K3 surface with infinitely many smooth rational curves. Precisely, suppose $X$ has an elliptic fibration with a section $C$ of infinite order. Then the multiples $nC$ (wrt addition law on the elliptic curve) are also sections, thus smooth rational curves. Now, being sections, they must be disjoint. But this is what seems puzzling to me: I thought that two disjoint smooth rational curves on a K3 surface should give independent classes in the Neron-Severi group, and so how come there can be infinitely many while $\rho(X)$ is bounded? I guess I am missing something.



Elliptic K3 surface with a section of infinite order
(I am just posting my comment as an answer at the OP's request.)


To make Sergey's answer even more concrete, try an example such as the Fermat quartic in $\mathbf P^3$. Here an elliptic fibration is given by projecting away from a line on the surface; any other line that is disjoint from the projection centre will give a section. But you can easily find two such lines that are not disjoint from each other. 


