Runtime verification of temporal properties over timed sequences of observations is crucial in various applications within cyber-physical systems ranging from autonomous vehicles over smart grids to medical devices. In this paper, we are addressing the challenge of effectively predicting the failure or success of properties in a continuous real-time setting. Our approach allows predictions to exploit assumptions on the system being monitored and supports predictions of non-observable system behaviour (e.g. internal faults). More concretely, in our approach properties are expressed in Metric Interval Temporal Logic (MITL), assumptions on the monitored system are specified in terms of Timed Automata, and observations are to be provided in terms of sequences of timed constraints. We present an assumption-based runtime verification algorithm and its implementation on top of the real-time verification tool UPPAAL. We show experimentally that assumptions can be effective in anticipating the satisfaction/violation of timed properties and in handling monitoring properties that predicate over unobservable events.