  As it is well known, the isomorphism problem for vertex-colored graphs with
color multiplicity at most 3 is solvable by the classical 2-dimensional
Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent
Cai-F\"urer-Immerman construction shows that even the multidimensional version
of the algorithm does not suffice for graphs with color multiplicity 4. We give
an efficient decision procedure that, given a graph $G$ of color multiplicity
4, recognizes whether or not $G$ is identifiable by 2-WL, that is, whether or
not 2-WL distinguishes $G$ from any non-isomorphic graph. In fact, we solve the
much more general problem of recognizing whether or not a given coherent
configuration of maximum fiber size 4 is separable. This extends our
recognition algorithm to graphs of color multiplicity 4 with directed and
colored edges.
  Our decision procedure is based on an explicit description of the class of
graphs with color multiplicity 4 that are not identifiable by 2-WL. The
Cai-F\"urer-Immerman graphs of color multiplicity 4 distinctly appear here as a
natural subclass, which demonstrates that the Cai-F\"urer-Immerman construction
is not ad hoc. Our classification reveals also other types of graphs that are
hard for 2-WL. One of them arises from patterns known as $(n_3)$-configurations
in incidence geometry.
