  Tian initiated the study of incomplete K\"ahler-Einstein metrics on
quasi-projective varieties with cone-edge type singularities along a divisor,
described by the cone-angle $2\pi(1-\alpha)$ for $\alpha\in (0, 1)$. In this
paper we study how the existence of such K\"ahler-Einstein metrics depends on
$\alpha$. We show that in the negative scalar curvature case, if such
K\"ahler-Einstein metrics exist for all small cone-angles then they exist for
every $\alpha\in(\frac{n+1}{n+2}, 1)$, where $n$ is the dimension. We also give
a characterization of the pairs that admit negatively curved cone-edge
K\"ahler-Einstein metrics with cone angle close to $2\pi$. Again if these
metrics exist for all cone-angles close to $2\pi$, then they exist in a uniform
interval of angles depending on the dimension only. Finally, we show how in the
positive scalar curvature case the existence of such uniform bounds is
obstructed.
