To ensure reliable communication in randomly varying and error-prone channels, wireless systems use adaptive modulation and coding (AMC) as well as hybrid ARQ (HARQ). In order to elucidate their compatibility and interaction, we compare the throughput provided by AMC, HARQ, and their combination (AMC-HARQ) under two operational conditions: in slow- and fast block-fading channels. Considering both, incremental redundancy HARQ (HARQ-IR) and repetition redundancy HARQ (HARQ-RR) we optimize the rate-decision regions for AMC/HARQ and compare them in terms of attainable throughput. Under a fairly general model of the channel variation and the decoding functions, we conclude that i) adding HARQ on top of AMC may be counterproductive in the high average signal- to-noise ratio regime for fast fading channels, and ii) HARQ is useful for slow fading channels, but it provides moderate throughput gains. We provide explanations for these results which allow us to propose paths to improve AMC-HARQ systems.