The performance of promptable video object segmentation (PVOS) models substantially degrades under input corruptions, which prevents PVOS deployment in safety-critical domains. This paper offers the first comprehensive study on robust PVOS (RobustPVOS). We first construct a new, comprehensive benchmark with two real-world evaluation datasets of 351 video clips and more than 2,500 object masks under real-world adverse conditions. At the same time, we generate synthetic training data by applying diverse and temporally varying corruptions to existing VOS datasets. Moreover, we present a new RobustPVOS method, dubbed Memory-object-conditioned Gated-rank Adaptation (MoGA). The key to successfully performing RobustPVOS is two-fold: effectively handling object-specific degradation and ensuring temporal consistency in predictions. MoGA leverages object-specific representations maintained in memory across frames to condition the robustification process, which allows the model to handle each tracked object differently in a temporally consistent way. Extensive experiments on our benchmark validate MoGA's efficacy, showing consistent and significant improvements across diverse corruption types on both synthetic and real-world datasets, establishing a strong baseline for future RobustPVOS research. Our benchmark is publicly available at https://sohyun-l.github.io/RobustPVOS_project_page/.