Since the 2008 financial crisis Government bond yields in US, Europe and elsewhere have been historically low and challenged term structure models that cannot rule out negative yields. This paper uses US and German Government yields to test three factor Gaussian models that do and that do not rule out negative yields, namely affine models, quadratic models, extensions of the Black and Black–Karasinski models. Quadratic models and a Vasicek-type model best fit observed yields when the stochastic factors driving the short rate are correlated. However the Black–Karasinski model for the US and the Black model for both US and Germany can best fit yields when interest rates are lowest, i.e. after 2008, despite the restriction of independent factors driving the short rate. A new linear-quadratic model whereby the central tendency of the short rate is a non-negative quadratic function of Gaussian factors performs particularly well for German yields. All models fit German yields better than US yields. All models fit the one year yield worse than longer term yields.