This paper provides an account of a new approach for system identification for time-varying systems. We consider the situation where the coherence time is significant relative to the system time-scale/memory/delay spread. Under these conditions it is not meaningful to track the system dynamics exactly. We address the problem by deliberately sacrificing performance by undermodeling the system with lower complexity models and gain in terms of guaranteeing accuracy. We quantify the estimation error as a function of the rate of variation, complexity of the model class and the undermodeling error. These bounds lead us to present fundamental limits to identifiability.