The variability of radio sources whose emission is dominated by single unresolved cores (core-dominated sources) is well estab-lished1. Evidence is also mounting that sources such as Cen A and 3C111, which are not core-dominated, also have varying cores2-4. Statistical studies of source fluxes using single-dish data and comparisons with known morphologies suggest that even in sources whose emission is swamped by that of the extended emission, the cores can be highly variable on timescales of weeks to years5,6. Here we present direct observational evidence for core variability of two lobe-dominated (weak-core) radio sources, GT1945 + 241 and GT0304 + 575. In the context of the beaming model, weak-core sources have little Doppler boosting and any residual variability is mainly intrinsic. The extreme nature of the core variability of the two sources described here suggests that the intrinsic variability can dominate beaming-related variability. If future studies reveal that this type of source is common, aspects of the beaming model, particularly in the context of the Unified Scheme, would be brought into question.