In conductor-insulator nanocomposites in which conducting fillers are dispersed in an insulating matrix, the electrical connectedness is established by inter-particle tunneling or hopping processes. These systems are intrinsically non-percolative and a coherent description of the functional dependence of the conductivity sigma on the filler properties, and in particular of the conductor-insulator transition, requires going beyond the usual continuum percolation approach by relaxing the constraint of a fixed connectivity distance. In this article, we consider dispersions of conducting spherical particles which are connected to all others by tunneling conductances and which are subjected to an effective attractive square-well potential. We show that the conductor-insulator transition at low contents phi of the conducting fillers does not determine the behavior of sigma at larger concentrations, in striking contrast to what is predicted by percolation theory. In particular, we find that at low phi the conductivity is governed almost entirely by the stickiness of the attraction, while at larger phi values sigma depends mainly on the depth of the potential well. As a consequence, by varying the range and depth of the potential while keeping the stickiness fixed, composites with similar conductor-insulator transitions may display conductivity variations of several orders of magnitude at intermediate and large phi values. By using a recently developed effective medium theory and the critical path approximation, we explain this behavior in terms of dominant tunneling processes which involve inter-particle distances spanning different regions of the square-well fluid structure as phi is varied. Our predictions could be tested in experiments by changing the potential profile with different depletants in polymer nanocomposites. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4705307]