Spectrum prediction is a key enabler of a range of emerging applications, from adaptive spectrum sensing to agile and proactive decision making, for dynamic spectrum access. Due to the fact that spectrum sensors easily experience an issue of missing readings and anomaly pollution, spectrum prediction with incomplete and corrupted historical observations has caused extensive concern. In this paper, we aim to tackle the challenging problem on how to accurately and efficiently recover the missing values from corrupted historical spectrum observations for approaching the limit of predictability in the spectrum prediction tasks. Specially, we first formulate a hankelized time-structured spectrum tensor that can naturally preserve both spectral and temporal dependencies among the historical spectrum observations. We then model the spectrum data recovery problem as tensor completion with its latent low-rank structure and sparse anomaly property. To efficiently solve the optimization problem, we design a robust online spectrum data recovery algorithm based on the alternating direction method. In addition, we also introduce the concept of maximum predictability to reveal the harmful effects of missing data and anomalies, as well as to evaluate the effectiveness of our proposed algorithm from an information theory perspective. Extensive experimental evaluations on the real-world spectrum observation dataset show that the proposed algorithm achieves significantly better spectrum data recovery performance in terms of both recovery accuracy and computation efficiency. It also indicates that not only the predictability of the frequency bands but also the prediction accuracy of any predictive algorithm can be improved by the proposed algorithm.