As an alternative to current wired-based networks, wireless sensor networks (WSNs) are becoming an increasingly compelling platform for engineering structural health monitoring (SHM) due to relatively low-cost, easy installation, and so forth. However, there is still an unaddressed challenge: the application-specific dependability in terms of sensor fault detection and tolerance. The dependability is also affected by a reduction on the quality of monitoring when mitigating WSN constrains (e.g., limited energy, narrow bandwidth). We address these by designing a dependable distributed WSN framework for SHM (called DependSHM) and then examining its ability to cope with sensor faults and constraints. We find evidence that faulty sensors can corrupt results of a health event (e.g., damage) in a structural system without being detected. More specifically, we bring attention to an undiscovered yet interesting fact, i.e., the real measured signals introduced by one or more faulty sensors may cause an undamaged location to be identified as damaged (false positive) or a damaged location as undamaged (false negative) diagnosis. This can be caused by faults in sensor bonding, precision degradation, amplification gain, bias, drift, noise, and so forth. In DependSHM, we present a distributed automated algorithm to detect such types of faults, and we offer an online signal reconstruction algorithm to recover from the wrong diagnosis. Through comprehensive simulations and a WSN prototype system implementation, we evaluate the effectiveness of DependSHM.