Many applications are implemented as multi-tier software systems, and are executed on distributed infrastructures, like cloud infrastructures, to benefit from the cost reduction that derives from dynamically allocating resources on-demand. In these systems, failures are becoming the norm rather than the exception, and predicting their occurrence, as well as locating the responsible faults, are essential enablers of preventive and corrective actions that can mitigate the impact of failures, and significantly improve the dependability of the systems. Current failure prediction approaches suffer either from false positives or limited accuracy, and do not produce enough information to effectively locate the responsible faults. In this paper, we present PreMiSE, a lightweight and precise approach to predict failures and locate the corresponding faults in multi-tier distributed systems. PreMiSE blends anomaly-based and signature-based techniques to identify multi-tier failures that impact on performance indicators, with high precision and low false positive rate. The experimental results that we obtained on a Cloud-based IP Multimedia Subsystem indicate that PreMiSE can indeed predict and locate possible failure occurrences with high precision and low overhead.