The title of the question is also the title of a talk by Vladimir Voevodsky, available here.  

Had this kind of opinion been expressed before?

EDIT. Thanks to all answerers, commentators, voters, and viewers! --- Here are three more links: 

Question arising from Voevodsky's talk on inconsistency by John Stillwell, 

Nelson's program to show inconsistency of ZF, by Andreas Thom, 

Pierre Colmez, La logique c’est pas logique !

EDIT. Here the link to the FOM list discussing these themes.  

The talk in question was given as part of a celebration (this past weekend) of the 80th anniversary of the founding of the Institute for Advanced Study in Princeton. As you might guess there were quite a few very well-known mathematicians and physicists in the audience. (To name just a few, Jack Milnor, Jean Bourgain, Robert Langlands, Frank Wilczek, and Freeman Dyson, all of whom also spoke during the weekend.)  The talk was a gem, and what did come as a surprise, at least to me, was that towards the end of his talk Voevodsky let on that he hoped that someone did find an inconsistency---and that by that time there was no audible gasp from the audience. There was of course a very lively discussion after the talk, and nobody seemed willing to say they felt that the "Current Foundations" (whatever they are) are definitely consistent. Of course Voevodsky was NOT saying that he felt that the body of theorems making up the "classic mathematics" that we normally deal with  might be inconsistent, that is quite a different matter.  What we should keep in mind is that a hundred years ago an earlier generation of mathematicians were quite surprised by not one but several "antinomies", like Russell's Paradox, The Burali-Forti Paradox, etc., (and that was followed by the greatest century in the history of Mathematics).  As to the question "Had this kind of opinion been expressed before?", yes of course it has, but perhaps not so forcefully or in such a high-level forum. One person who has been expressing such ideas in recent years is my old friend Ed Nelson, who was also in the audience. (You can see his ideas in a recent paper: http://www.math.princeton.edu/~nelson/papers/warn.pdf). I spoke  with him after the talk and he seemed pleased that it was now becoming acceptable to discuss the matter seriously.