Wo makes an interesting point: if someones believes in strong necessities despite appropriately conceivable counterexamples, on what grounds do they reject the idea of "strong possibilities", i.e. contradictions or inconceivabilities that are nonetheless held to be metaphysically possible?
In fact, I used to be pretty sympathetic to such a view. If modality isn't tied to the rational sphere, then it's hard to see how we could say anything much about it at all! (Though I recently took a shot at reconciling a deeply "realist" conception of modality with the epistemic strength of the modal rationalist, here.)