  Given a finite root system $\Phi$, we show that 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 $L,L'$.
There also 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, a
similar result holds 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, sufficient to give similar results for
algebraic and generic cohomology. The results both use, and have consequences
for, Kazhdan-Lusztig polynomials. An appendix proves a stable version, needed
for small prime arguments, of Donkin's tilting module conjecture.
