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Applications from finance to epidemiology and cyber-security require accurate forecasts of dynamic phenomena, which are often only partially observed. We demonstrate that a system's predictability degrades as a function of temporal sampling, regardless of the adopted forecasting model. We quantify the loss of predictability due to sampling, and show that it cannot be recovered by using external signals. We validate the generality of our theoretical findings in real-world partially observed systems representing infectious disease outbreaks, online discussions, and software development projects. On a variety of prediction tasks---forecasting new infections, the popularity of topics in online discussions, or interest in cryptocurrency projects---predictability irrecoverably decays as a function of sampling, unveiling fundamental predictability limits in partially observed systems.