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Federated learning (FL) enables edge devices to collaboratively learn a model in a distributed fashion. Many existing researches have focused on improving communication efficiency of high-dimensional models and addressing bias caused by local updates. However, most of FL algorithms are either based on reliable communications or assume fixed and known unreliability characteristics. In practice, networks could suffer from dynamic channel conditions and non-deterministic disruptions, with time-varying and unknown characteristics. To this end, in this paper we propose a sparsity enabled FL framework with both communication efficiency and bias reduction, termed as SAFARI. It makes novel use of a similarity among client models to rectify and compensate for bias that is resulted from unreliable communications. More precisely, sparse learning is implemented on local clients to mitigate communication overhead, while to cope with unreliable communications, a similarity-based compensation method is proposed to provide surrogates for missing model updates. We analyze SAFARI under bounded dissimilarity and with respect to sparse models. It is demonstrated that SAFARI under unreliable communications is guaranteed to converge at the same rate as the standard FedAvg with perfect communications. Implementations and evaluations on CIFAR-10 dataset validate the effectiveness of SAFARI by showing that it can achieve the same convergence speed and accuracy as FedAvg with perfect communications, with up to 80% of the model weights being pruned and a high percentage of client updates missing in each round.