This technical note studies the problem of distributed fusion estimation for a class of networked multi-sensor fusion systems (NMFSs) with limited communication capacity and affected by sensor noises and disturbances. A novel data compression strategy is proposed to reduce communication cost, which leads to a stochastic model describing the communication capacity. To obtain an optimal weighting fusion criterion, the distributed mixed H-2/H-infinity fusion estimation problem is converted into a convex optimization problem by using Lyapunov theory and matrix analysis approach, which can be easily solved by standard software packages. The obtained fusion estimator gains are time-invariant, which will not increase computation cost of the fusion center. An illustrative example is given to show the effectiveness of the proposed methods.