It has been widely observed that exact or approximate MAP (mode-seeking) decoding from natural language generation (NLG) models consistently leads to degenerate outputs (Holtzman et al., 2019; Stahlberg and Byrne, 2019). Prior work has attributed this behavior to either a fundamental and unavoidable inadequacy of modes in probabilistic models or weaknesses in language modeling. Contrastingly, we argue that degenerate modes can even occur in the absence of any modeling error, due to contamination of the training data. Specifically, we argue that mixing even a tiny amount of low-entropy noise with a population text distribution can cause the data distribution's mode to become degenerate. We therefore propose to apply MAP decoding to the model's true conditional distribution where the conditioning variable explicitly avoids specific degenerate behavior. Using exact search, we empirically verify that the length-conditional modes of machine translation models and language models are indeed more fluent and topical than their unconditional modes. For the first time, we also share many examples of exact modal sequences from these models, and from several variants of the LLaMA-7B model. Notably, we observe that various kinds of degenerate modes persist, even at the scale of LLaMA-7B. Although we cannot tractably address these degeneracies with exact search, we perform a classifier-based approximate search on LLaMA-7B, a model which was not trained for instruction following, and find that we are able to elicit reasonable outputs without any finetuning.