The Hubbert-Deffeyes "peak oil" (HDPO) model predicts that world oil production is about to enter a period of sustained decline. This paper investigates the empirical robustness of this claim. I use out-of-sample methods to test whether the HDPO model is capable of estimating ultimately recoverable reserves. HDPO model estimates of ultimately recoverable reserves, based on data available 30 years or more in the past, are found to be less than current observed cumulative production and discoveries. This result is robust to different specifications of the HDPO model, to applications to production and discoveries data, and to various levels of geographical aggregation. These problems stem from an attempt by the HDPO model to force a linear relationship onto data which are inherently non-linear This characteristic of the data is present in a wide variety of natural resources. I also show that the HDPO model is incapable of distinguishing between processes for which cumulative production is truly finite and processes for which cumulative production is unbounded. These findings undermine claims that the HDPO model is capable of yielding meaningful measures of ultimately recoverable reserves or of predicting when world oil production might peak.