Any smooth manifold $M$ admits a metric with quadratic curvature decay. 
In fact there is a complete Riemannian metric $g$ on $M$ such that $$|K_x|=o(|x_0x|^{-2}),$$
here $|K_x|$ is absolute bound for sectional curvature of $g$ at point $x$,
$x_0$ is a fixed point and and $|x_0x|$ denotes distance induced by $g$.

In particular there is no examples which you are looking for.

See p. 96 in Gromov M., Volume and bounded cohomology, Publ. Math. IHES 56 (1982) 5–99.