In this paper, we analyze the predictability of the movements of bond premia of US Treasury due to oil price uncertainty over the monthly period 1953:06 to 2016:12. For our purpose, we use a higher order nonparametric causality-in-quantiles framework, which in turn, allows us to test for predictability over the entire conditional distribution of not only bond returns, but also its volatility, by controlling for misspecification due to uncaptured nonlinearity and structural breaks, which we show to exist in our data. We find that oil uncertainty not only predicts (increases) US bond returns, but also its volatility, with the effect on the latter being stronger. In addition, oil uncertainty tends to have a stronger impact on the shortest and longest maturities (2-and 5-year), and relatively weaker impact on bonds with medium-term (3-and 4-year) maturities. Our results are robust to alternative measures of oil market uncertainty and bond market volatility.