There is overwhelming consensus that the intensity of heavy precipitation events is increasing in a warming world. It is generally expected such increases will translate to a corresponding increase in flooding. Here, using global data sets for non-urban catchments, we investigate the sensitivity of extreme daily precipitation and streamflow to changes in daily temperature. We find little evidence to suggest that increases in heavy rainfall events at higher temperatures result in similar increases in streamflow, with most regions throughout the world showing decreased streamflow with higher temperatures. To understand why this is the case, we assess the impact of the size of the catchment and the rarity of the event. As the precipitation event becomes more extreme and the catchment size becomes smaller, characteristics such as the initial moisture in the catchment become less relevant, leading to a more consistent response of precipitation and streamflow extremes to temperature increase. Our results indicate that only in the most extreme cases, for smaller catchments, do increases in precipitation at higher temperatures correspond to increases in streamflow.