Auto-scaling is crucial for achieving elasticity in cloud databases as well as other cloud systems. Predictive auto-scaling, which leverages forecasting techniques to adjust resources based on predicted workload, has been widely adopted. However, the inherent inaccuracy of forecasting presents a significant challenge, potentially causing resource under-provisioning. To address this challenge, we propose robust predictive auto-scaling that considers the uncertainty in forecasts. Unlike previous predictive approaches that rely on single-valued forecasts, we leverage probabilistic forecasting techniques to generate quan-tile forecasts, providing a more comprehensive understanding of the potential future workloads. By formulating the auto-scaling problem as a robust optimization problem, we enable the implementation of auto-scaling strategies with customizable levels of robustness, which can be determined by considering various quantile levels of forecasts. Moreover, we enhance the adaptability of our strategy by incorporating different quantile levels through-out the entire decision horizon, allowing for dynamic adjustments in the conservatism of our auto-scaling decisions. This enables us to strike a balance between resource efficiency and system robustness. Through extensive experiments, we demonstrate the effectiveness of our approach in achieving robust auto-scaling in cloud databases, while maintaining reasonable resource efficiency.