In Natural Language Generation (NLG) tasks, for any input, multiple communicative goals are plausible, and any goal can be put into words, or produced, in multiple ways. We characterise the extent to which human production varies lexically, syntactically, and semantically across four NLG tasks, connecting human production variability to aleatoric or data uncertainty. We then inspect the space of output strings shaped by a generation system's predicted probability distribution and decoding algorithm to probe its uncertainty. For each test input, we measure the generator's calibration to human production variability. Following this instance-level approach, we analyse NLG models and decoding strategies, demonstrating that probing a generator with multiple samples and, when possible, multiple references, provides the level of detail necessary to gain understanding of a model's representation of uncertainty. Code available at https://github.com/dmg-illc/nlg-uncertainty-probes.