Using a century of data, we show that Treasury convenience yield and inflation comove positively during the inflationary 1970s-1980s, but negatively pre-WWII and post-2000. An inflation decomposition reveals that higher supply inflation predicts higher convenience, while lower demand inflation follows higher convenience. In our model, inflationary cost-push shocks raise the opportunity cost of holding money and money-like assets, inducing higher convenience, as in 1970s-1980s. Conversely, liquidity demand shocks drive up convenience but lower consumption demand and inflation in the model, as pre-WWII and post-2000. By linking the evidence to macroeconomic drivers, our results challenge the notion that inflation directly depresses Treasury convenience.