Skillful forecasts of extreme weather events have a major socioeconomic relevance. Here, we compare two complementary approaches to diagnose the predictability of extreme weather: recent developments in dynamical systems theory and numerical ensemble weather forecasts. The former allows us to define atmospheric configurations in terms of their persistence and local dimension, which provides information on how the atmosphere evolves to and from a given state of interest. These metrics may be used as proxies for the intrinsic predictability of the atmosphere, which only depends on the atmosphere's properties. Ensemble weather forecasts provide information on the practical predictability of the atmosphere, which partly depends on the performance of the numerical model used. We focus on heat waves affecting the eastern Mediterranean. These are identified using the climatic stress index (CSI), which was explicitly developed for the summer weather conditions in this region and differentiates between heat waves (upper decile) and cool days (lower decile). Significant differences are found between the two groups from both the dynamical systems and the numerical weather prediction perspectives. Specifically, heat waves show relatively stable flow characteristics (high intrinsic predictability) but comparatively low practical predictability (large model spread and error). For 500 hPa geopotential height fields, the intrinsic predictability of heat waves is lowest at the event's onset and decay. We relate these results to the physical processes governing eastern Mediterranean summer heat waves: adiabatic descent of the air parcels over the region and the geographical origin of the air parcels over land prior to the onset of a heat wave. A detailed analysis of the mid-August 2010 record-breaking heat wave provides further insights into the range of different regional atmospheric configurations conducive to heat waves. We conclude that the dynamical systems approach can be a useful complement to conventional numerical forecasts for understanding the dynamics and predictability of eastern Mediterranean heat waves.