We know theoretically that language in general is non-context-free. However, proven examples of non-context-free-ness are rare. The field has not really moved away from CFGs as the basis for formalisms (otherwise we can't really parse anything), but the complexity of attributes that need to be passed around to account for even basic phenomena (agreement, question or relative-clause formation, etc.) is such that CFG trees become very complex. Instead people (HPSG and CCG people, for instance) now prefer to use lexicalist models with everything stuffed inside lexical units with complex bottom-up derivation/unification rules.