Let
G
G
be an unmixed bipartite graph of dimension
d
−
1
d-1
. Assume that
K
n
,
n
K_{n,n}
, with
n
≥
2
n\ge 2
, is a maximal complete bipartite subgraph of
G
G
of minimum dimension. Then
G
G
is Cohen-Macaulay in codimension
t
t
if and only if
t
≥
d
−
n
+
1
t\ge d-n+1
. This is derived from a characterization of Cohen-Macaulay bipartite graphs by Herzog and Hibi and generalizes a recent result of Cook and Nagel on unmixed Buchsbaum graphs. Furthermore, we show that any unmixed bipartite graph
G
G
which is Cohen-Macaulay in codimension
t
t
, is obtained from a Cohen-Macaulay graph by replacing certain edges of
G
G
with complete bipartite graphs. Thus, in light of combinatorial characterization of Cohen-Macaulay bipartite graphs, our result may be considered purely combinatorial.