In natural and urban canyon environments, Global Navigation Satellite System (GNSS) signals suffer from various challenges such as signal multipath, limited or lack of signal availability and poor geometry. Inertial sensors are often employed to improve the solution continuity under poor GNSS signal quality and availability conditions. Various fault detection schemes have been proposed in the literature to detect and remove biased GNSS measurements to obtain a more reliable navigation solution. However, many of these methods are found to be sub-optimal and often lead to unavailability of reliability measures, mostly because of the improper characterization of the measurement errors. A robust filtering architecture is thus proposed which assumes a heavy-tailed distribution for the measurement errors. Moreover, the proposed filter is capable of adapting to the changing GNSS signal conditions such as when moving from open sky conditions to deep canyons. Results obtained by processing data collected in various GNSS challenged environments show that the proposed scheme provides a robust navigation solution without having to excessively reject usable measurements. The tests reported herein show improvements of nearly 15% and 80% for position accuracy and reliability, respectively, when applying the above approach.