Petroni et al. (2019) demonstrated that it is possible to retrieve world facts from a pre-trained language model by expressing them as cloze-style prompts and interpret the model’s prediction accuracy as a lower bound on the amount of factual information it encodes. Subsequent work has attempted to tighten the estimate by searching for better prompts, using a disjoint set of facts as training data. In this work, we make two complementary contributions to better understand these factual probing techniques. First, we propose OPTIPROMPT, a novel and efficient method which directly optimizes in continuous embedding space. We find this simple method is able to predict an additional 6.4% of facts in the LAMA benchmark. Second, we raise a more important question: Can we really interpret these probing results as a lower bound? Is it possible that these prompt-search methods learn from the training data too? We find, somewhat surprisingly, that the training data used by these methods contains certain regularities of the underlying fact distribution, and all the existing prompt methods, including ours, are able to exploit them for better fact prediction. We conduct a set of control experiments to disentangle “learning” from “learning to recall”, providing a more detailed picture of what different prompts can reveal about pre-trained language models.