The problem of determining, given a string $x$ and a context-free grammar $G$, whether $x \in L(G)$ is conjectured to take more than linear time in the length of $x$. Currently the best known algorithm is Valiant's which takes $\Theta(|G||x|^\omega)$.
On the other hand, for a fixed $G$ there may be a specialized recognition algorithm which is faster, and indeed linear-time algorithms have been developed for many grammars and classes of grammars of interest. In fact, I can't think of any example of a language which doesn't have a linear-time recognition algorithm.
Is there any example of a context-free language that is not known to be recognizable in linear time?