This paper deals with error bound characterizations of the conical constraint qualification condition for a convex inequality system in a Banach space X. We establish necessary and sufficient conditions for a closed convex set S defined by a convex function g to have the conical constraint qualification. These conditions are expressed in terms of the notion of error bound. Our results show that these characterizations hold in the following special cases: 1. g is the maximum of a finite number of differentiable convex functions. 2. S is closed convex and polyhedral. 3. The dimension of the subspace lin(S) is less than 2 and g is positively homogeneous. We construct technical examples showing that these characterizations are limited to the three situations above. We introduce a new condition in terms of the gauge function which allows us to give an error bound characterization of convex nondifferentiable systems and to obtain as a direct consequence different characterizations of the concept of strong conical hull intersection property (CHIP) for a finite collection of convex sets.