Here, we show that the conductivity of conductor-insulator composites in which electrons can tunnel from each conducting particle to all others may display both percolation and tunneling (i.e., hopping) regimes depending on few characteristics of the composite. Specifically, we find that the relevant parameters that give rise to one regime or the other are D/xi (where D is the size of the conducting particles and xi is the tunneling length) and the specific composite microstructure. For large values of D/xi, percolation arises when the composite microstructure can be modeled as a regular lattice that is fractionally occupied by conducting particle while the tunneling regime is always obtained for equilibrium distributions of conducting particles in a continuum insulating matrix. As D/xi decreases the percolating behavior of the conductivity of latticelike composites gradually crosses over to the tunnelinglike regime characterizing particle dispersions in the continuum. For D/xi values lower than D/x similar or equal to 5 the conductivity has tunnelinglike behavior independent of the specific microstructure of the composite.