Quillen's Theorem A says that a functor between (small) categories $f:I\rightarrow J$ induces a weak equivalence of the nerves if for each $j\in J$ the comma category $f/j$ is weakly contractible. In practice, one frequently shows a category is weakly contractible by showing it has an initial or terminal object, or that it is (co)-filtering. In the examples that keep coming up, the only obvious property is that the comma categories are connected and admit pullbacks.
Is this enough to show that these categories are weakly contractible? If not, is there a good counterexample?