this isn't really true anymore. First, a number of the big results are constructions, not counterexamples. For example the existence of a non-sofic group. It was widely believed that non-sofic groups existed (so it wasn't a "counterexample" to a widely believed conjecture), but no constructions were known.

There are other examples though. For example, NP hardness of n^{1/400}-approx CVP. Like any NP hardness proof, this shows you can faithfully encode a hard problem (3SAT here iirc) in terms of another candidate hard problem. Not really a counterexample at all.