Measurements provided by the phasor measurement units (PMUs) in a power network can be highly erroneous. Furthermore, some generating units may not have a local PMU, therefore it may not be possible to obtain high accurate and reliable results based on the previously studied dynamic state estimation approaches, which rely on the raw measurements provided by the PMUs. In order to address these issues, this paper presents a robust distributed dynamic state estimation approach that not only is robust against bad data, but also makes it possible to obtain the dynamic state estimation results for the generators without a local PMU. The proposed approach also accounts for expected delays in receiving estimated measurements by using a multi-step ahead state predictor to correct for delayed inputs. This procedure can also be useful for the short-term transient stability predictions.