Most of time-varying systems in fusion estimation are generally modeled without bounded noises. In this paper, we study the distributed fusion estimation problem for networked time-varying systems with bounded noises, where the resource constraints (i.e., bandwidth or energy) and quantization effect are described by a unified model. A new local estimator with time-varying gain is designed by solving a class of convex optimization problems such that the square error of the estimator is bounded. When each local estimate is transmitted to the fusion center over communication networks, the selection probability criterion is derived such that the mean square error of the each compensating state estimate is bounded. Then, a convex optimization problem on the design of an optimal weighting fusion criterion is established in terms of linear matrix inequalities, which can be solved by standard software packages. Target tracking system with time-varying sampling period is given to show the effectiveness of the proposed method.