  A Riemannian manifold is called almost positively curved if the set of points
for which all $2$-planes have positive sectional curvature is open and dense.
We find three new examples of almost positively curved manifolds:
$Sp(3)/Sp(1)^2$, and two circle quotients of $Sp(3)/Sp(1)^2$. We also show the
quasi-positively curved metric of Tapp [26]} on $Sp(n+1)/Sp(n-1) Sp(1)$ is not
almost positively curved if $n\geq 3$.
