Cloud platforms have been focusing on reducing their carbon emissions by shifting workloads across time and locations to when and where low-carbon energy is available. Despite the prominence of this idea, prior work has only quantified the potential of spatiotemporal workload shifting in narrow settings, i.e., for specific workloads in select regions. In particular, there has been limited work on quantifying an upper bound on the ideal and practical benefits of carbon-aware spatiotemporal workload shifting for a wide range of cloud workloads. To address the problem, we conduct a detailed data-driven analysis to understand the benefits and limitations of carbon-aware spatiotemporal scheduling for cloud workloads. We utilize carbon intensity data from 123 regions, encompassing most major cloud sites, to analyze two broad classes of workloads -- batch and interactive -- and their various characteristics, e.g., job duration, deadlines, and SLOs. Our findings show that while spatiotemporal workload shifting can reduce workloads' carbon emissions, the practical upper bounds of these carbon reductions are currently limited and far from ideal. We also show that simple scheduling policies often yield most of these reductions, with more sophisticated techniques yielding little additional benefit. Notably, we also find that the benefit of carbon-aware workload scheduling relative to carbon-agnostic scheduling will decrease as the energy supply becomes "greener".