In the standard AGN reverberation-mapping model, variations in broad-line region (BLR) fluxes can be predicted from variations in the optical continuum (taken as a proxy for variations in the ionizing continuum). However, it has long been known that BLR variability can deviate from these predictions. Extensive, long-term Hβ and continuum monitoring of NGC 5548 and a large sample of other high-quality Hβ light curves of AGNs are analyzed to investigate the characteristics and frequency of anomalous responses of the BLR. We find that anomalies are common and most likely occur in every object. Onsets occur on a timescale slightly longer than the light-crossing time and durations are of the order of the characteristic timescale of variability of the optical continuum to several times longer. The only correlation with continuum variability is that anomalies are larger when NGC 5548 is in a low state. Evidence that the optical continuum of AGNs can vary independently from the higher-energy continua is sufficient to explain anomalous responses. It is also reasonable to believe that the frequent lack of correlation between different spectral regions is due to anisotropic and non-axisymmetric emission, as well as rapid changes in line profiles, velocity-dependent time lags, or movement of absorbing clouds across the line of sight. Reverberation mapping can be used to determine AGN mass, but relies on the assumption that the ionizing continuum varies in response to continuum variability. Therefore, no matter the cause of these anomalies, their prevalence must be accounted for when planning reverberation-mapping campaigns.