I don't personally believe that CFL are insufficient, but among linguists who care about weak generative capacity (probably most don't care about the issue), the consensus seems to be that they are. The usual reference given is an article by Peter Shieber, which is online here, and which is, in my opinion, an admirable work of scholarship. (Which doesn't make it right, of course.)
So far as I know, the case rests on the grammaticality of unbounded cross-serial dependencies in certain natural languages, see Cross-serial dependencies. Or, one might refer to them as "respectively constructions".
A conceptual example would be "John smoked, ate, and drank cigars, caviar, and brandy, respectively", supposing that is interpreted as meaning "John smoked cigars, John ate caviar, and John drank brandy". The problem is getting the verbs to pair up with the objects that they go with. What seems to be a corresponding problem with CFL is generating "copy" languages, which have sentences with two consecutive copies of the same string, which is provably impossible in general.
So far (you're probably thinking) this seems to be an imaginary problem, since the English example I gave above is not actually grammatical. If by proposing CFL for natural languages, we predict that natural languages cannot have such constructions, that would seem to be a good thing.
The problem is that some other languages do seem to have such constructions. Shieber described a Swiss-German dialect, and others have since been found. So even if it happens that English doesn't have cross-serial constructions, since we're interested in human language in general, we have to give up on CFL as a general model.
I don't, myself, think this argument has any force, since the only real prediction that CFL makes is that the depth of cross-serial constructions must be bounded and that the greater the depth of the cross-serial construction the more complex the CFL must be to emulate it. That is the situation in Swiss-German, in Shieber's account. Two-depth CS constructions are accepted by all, three-depth ones by some, but four-depth ones are not accepted by anyone. So, in my view, the facts actually support the theory that natural languages are context free, rather than showing that they are not.
On your last question, yes, quite a few people believe that natural language is r.e., recursively enumerable, i.e., since all the languages in the Chomsky hierarchy are r.e. One of the dissidents, I believe, would be C. F. Hockett, who proposed in The State of the Art, reviewed here by Chuck Fillmore, that human languages do not have definite systems of the sort that Chomsky assumes.