Safety-critical navigation applications require that estimation errors be reliably quantified and bounded. Over the last decade, significant effort has been put to guarantee a bounded position estimation by using Global Navigation Satellite Systems (GNSS) in satellite-based or ground-based augmentation systems (SBAS, GBAS) and Advanced Receiver Autonomous Integrity Monitoring (ARAIM) for aviation. This has been achieved by carefully designing models that overbound the different residual error components in range measurements (e.g., satellite clock and orbit, tropospheric and multipath among others). On the other hand, and as part of Avionics-based Augmentation Systems (ABAS), the use of Inertial Reference Systems (IRS) has been traditionally included as additional source of redundant navigation information. More recently, the use of Inertial Navigation Systems (INS) with a wider spectrum of possible inertial sensor qualities in tighter integration with GNSS has seen its way in a new Minimum Operational Performance Standard (MOPS). New GNSS/INS systems and standards could still benefit from the methodologies and aspects developed for future multifrequency-multiconstellation GNSS standards. However, safety-related GNSS systems like ARAIM are snapshot-based, that is, the position estimation is performed independently at every epoch, whereas GNSS/INS systems are typically based on Kalman filtering (KF). Therefore, the existing error overbounding models and methodologies are not enough to produce a robust KF position estimation since the impact of time-correlation must also be accounted for. Moreover, it has been observed that the time-correlation of different GNSS errors present also some level of uncertain behavior, which makes very challenging for linear dynamic systems to produce a guaranteed solution. As proposed by GNSS MOPS, there are sources of time-correlated errors that can be well modelled using Gauss-Markov processes (GMP). Using this GMP parametric model, it is possible to capture the uncertain time-correlated nature of errors processes by allowing the variance and time correlation constant of the GMP model to be in a bounded range. Under this situation, the first part of this thesis studies the propagation of the uncertain models through the Kalman filter estimation and provides new theoretical tools in time and frequency domain to bound the KF error estimation covariance. As a result, tight stationary bounding models on the GMP uncertain processes are derived in both continuous and discrete time domain. This is extended to non-stationary models that provide tighter error bounding during an initial transient phase when measurements are first introduced (which will be relevant in scenarios with changing number of visible satellites). The new models can very easily be used during the KF implementation which might be very attractive by regulators and designers. In the second part of the thesis, the new overbounding GMP models are applied for a tightly-coupled GNSS/INS integration. The design of the filter and error models are performed following compatibility with current aviation standards and ARAIM Working Group C results. The impact of the use of the new models is analysed in terms of conservativeness, integrity and availability based on realistic trajectory simulations. The benefit of an overbounded GNSS/INS solution is also compared with the current baseline ARAIM algorithm solution.