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Compressive data gathering, which is based on the recent breakthroughs in compressive sensing theory, has been proposed as a viable approach for sensor network data collection at low communication overhead. Nevertheless, it suffers from a low data recovery accuracy when outlying sensor readings and broken links exist. In this paper, we investigate the impact of outlying sensor readings and broken links on high-fidelity data gathering, and propose approaches based on the compressive sensing theory to identify outlying sensor readings and derive the corresponding accurate values, and to infer broken links. Our design is validated by a comparison based extensive simulation study, and the results indicate that compressive data gathering is superior over traditional in-network data compression techniques for practical sensor network settings.