Accurate and robust trajectory predictions of road users are needed to enable safe automated driving. To do this, machine learning models are often used, which can show erratic behavior when presented with previously unseen inputs. In this work, two environment-aware models (MotionCNN and MultiPath++) and two common baselines (Constant Velocity and an LSTM) are benchmarked for robustness against various perturbations that simulate functional insufficiencies observed during model deployment in a vehicle: unavailability of road information, late detections, and noise. Results show significant performance degradation under the presence of these perturbations, with errors increasing up to +1444.8\% in commonly used trajectory prediction evaluation metrics. Training the models with similar perturbations effectively reduces performance degradation, with error increases of up to +87.5\%. We argue that despite being an effective mitigation strategy, data augmentation through perturbations during training does not guarantee robustness towards unforeseen perturbations, since identification of all possible on-road complications is unfeasible. Furthermore, degrading the inputs sometimes leads to more accurate predictions, suggesting that the models are unable to learn the true relationships between the different elements in the data.