In this work we investigate the contribution of inter-fiber cohesion to defining the mechanical behavior of stochastic crosslinked fiber networks. Fibers are athermal and store energy primarily in their bending and axial deformation modes. Cohesion between fibers is defined by an interaction potential. These structures are in equilibrium with the inter-fiber cohesive forces before external load is applied and their mechanical behavior is probed in uniaxial tension. Two types of configurations are considered: a state with high initial free volume in which contacts between fibers are scarce, and a state with low free volume and large number of fiber contacts. While in the absence of cohesion the response is hyperelastic, we observe that a yield point-like phenomenon develops as the strength of cohesion increases in both network types considered; we refer to this as an 'unlocking phenomenon'. The small strain stiffness increases as cohesion becomes more pronounced. The stiffness and unlocking stress are expressed in terms of network parameters and cohesion strength through a product of two functions, one dependent on network parameters only, and the other is a function of the cohesion strength. While the small strain response is controlled by cohesion, the large strain behavior is shown to be largely controlled by the network. Therefore, varying the strength of cohesion has no effect on strain stiffening. These observations provide a physical basis for the unlocking observed in both athermal and thermal network materials and are expected to facilitate the design of soft materials with novel properties.