I've been wondering wether there is a bijection between the set of killing vector fields and the group of all isometries of a fixed given manifold. Is it true in general, when does it fail to happen? It's obvious from definition that the number of killing vector fields does not exceed that of all isometries of the manifold (this I can actually see! ;)
In those cases of affirmative answer, how's the bijection or, in other words, how's the counting process?
A Killing vector field generates an infinite family of isometries - for any $t \in \mathbb R$ we can flow along the field for time $t$. Also note that linear combinations of Killing fields are Killing; so we expect to have an infinite number of these as well. Thus just counting isn't very useful here.
The "correct" question to ask is about the dimension of the isometry group, in which case the answer is very direct: the Lie algebra of the isometry group is (isomorphic to) the space of Killing fields, so we can conclude that the dimension of the group is equal to the number of independent Killing fields. They don't tell you about the global structure of the group - for example the Killing vector fields of the sphere $S^n$ generate the orientation-preserving rotations $SO(n+1)$, but not reflections.