Weight thresholding is a simple technique that aims at reducing the number of edges in weighted networks that are otherwise too dense for the application of standard graph theoretical methods. We show that the group structure of real weighted networks is very robust under weight thresholding, as it is maintained even when most of the edges are removed. This appears to be related to the correlation between topology and weight that characterizes real networks. On the other hand, the behavior of other properties is generally system dependent.