We propose a model for leakage currents in Schottky contacts on InAlN/GaN heterostructures based on two distinct tunneling mechanisms. Our modeling relies on structural parameters, in particular, InAlN dielectric constant, interface polarization charges and Schottky barrier height, which are experimentally determined in the first part of our work. The first leakage mechanism is dominant in heterostructures with very thin (<= 7 nm) InAlN barriers and consists in tunneling assisted by a deep level located 1.7 eV below the InAlN conduction band edge. We provide experimental evidence for this level through photocapacitance measurements. The second mechanism is on the other hand dominant in thicker InAlN layers and is linked to the appearance of highly doped regions where direct tunneling through the whole InAlN barrier is significantly enhanced. We also show that the two mechanisms may coexist for InAlN layers of intermediate thickness. Our findings confirm a progressive degradation of the InAlN material quality as the layer is grown thicker. (C) 2014 AIP Publishing LLC.