In this paper the ubiquity of non-Buchsbaum but quasi-Buchsbaum rings is established. The result is stated as follows: Let
d
≥
3
d \ge 3
and
h
1
,
h
2
,
…
,
h
d
−
1
≥
0
{h_1},{h_2}, \ldots ,{h_{d - 1}} \ge 0
be integers and assume that at least two of
h
i
{h_i}
’s are positive. Then there exists a non-Buchsbaum but quasi-Buchsbaum local integral domain
A
A
of
dim
A
=
d
\dim A = d
and such that
l
A
(
H
m
i
(
A
)
)
=
h
i
{l_A}(H_m^i(A)) = {h_i}
for all
1
≤
i
≤
d
−
1
1 \le i \le d - 1
. Moreover if
h
1
=
0
{h_1} = 0
the ring
A
A
can be chosen to be normal.