This paper can be seen as a continuation of the works contained in the recent article (J. Alg., 305 (2006), 949–956) of the second author, and those of Juan Migliore (math. AC/0508067). Our results are: 1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property (WLP). In fact, for
e
≫
0
e\gg 0
, we will construct a codimension three, type two
h
h
-vector of socle degree
e
e
such that all the level algebras with that
h
h
-vector do not have the WLP. We will also describe the family of those algebras and compute its dimension, for each
e
≫
0
e\gg 0
. 2). There exist reduced level sets of points in
P
3
{\mathbf P}^3
of type two whose artinian reductions all fail to have the WLP. Indeed, the examples constructed here have the same
h
h
-vectors we mentioned in 1). 3). For any integer
r
≥
3
r\geq 3
, there exist non-unimodal monomial artinian level algebras of codimension
r
r
. As an immediate consequence of this result, we obtain another proof of the fact (first shown by Migliore in the above-mentioned preprint, Theorem 4.3) that, for any
r
≥
3
r\geq 3
, there exist reduced level sets of points in
P
r
{\mathbf P}^r
whose artinian reductions are non-unimodal.