Given a finite root system $\Phi$, we show there is an integer $c=c(\Phi)$
such that $\dim\Ext_G^1(L,L')<c$, for any reductive algebraic group $G$ with
root system $\Phi$ and any irreducible rational $G$-modules $L,L'$. We also
prove that there is such a bound in the case of finite groups of Lie type,
depending only on the root system and not on the underlying field. For quantum
groups, we are able to obtain a similar result for $\Ext^n$, for any integer
$n\geq 0$, using a constant depending only on $n$ and the root system. Weaker
versions of this are proved in the algebraic and finite group cases for large
characteristic. Our results both use, and have consequences for,
Kazhdan-Lusztig polynomials. We also introduce the new tool of a shifted
standard (or costandard) module filtration for an object in the derived
category of a highest weight category.