Network navigation is a promising paradigm for providing location awareness in wireless environments, where nodes estimate their locations based on sensor measurements and prior knowledge. In the presence of limited wireless resources, only a subset rather than all of the node pairs can perform inter-node measurements. The procedure of selecting node pairs at different times for inter-node measurements, referred to as network scheduling, affects the evolution of the localization errors. Thus, it is crucial to design efficient scheduling strategies for network navigation. This paper introduces situation-aware scheduling that exploits network states to select measurement pairs, and develops a framework to characterize the effects of scheduling strategies and of network settings on the error evolution. In particular, both sufficient and necessary conditions for the boundedness of the error evolution are provided. Furthermore, opportunistic and random situation-aware scheduling strategies are proposed, and bounds on the corresponding time-averaged network localization errors are derived. These strategies are shown to be optimal in terms of the error scaling with the number of agents. Finally, the reduction of the error scaling by increasing the number of simultaneous measurement pairs is quantified.