Gromov's random groups probably provide examples of the phenomenon you describe. The idea is to construct a group by fixing a number of generators and randomly choose relations in some specified way. A number of counter-examples to hard conjectures related to geometric group theory (such as the Baum-Connes conjecture with coefficients) have been shown to exsit recently by arguing that a random group is a counter-example with positive probability. But in many cases it is not known how to verify that a specific group (i.e. a specific set of relations) has the desired property.