Adapting J. T. Stafford’s [Invent. Math. 79, 619-638 (1985; Zbl 0558.17011)] construction of non-holonomic simple modules over the complex Weyl algebras \(A_ n\), \(n \geq 2\), the author constructs non-simple \(A_ n\)-modules of finite length that are also critical with respect to the Gelfand-Kirillov dimension. Thus, Tauvel’s question whether critical modules of finite length over a solvable Lie algebra are necessarily simple has a negative answer already for nilpotent Lie algebras. This is interesting in view of the fact that the answer to this question is positive for commutative polynomial rings. The paper also presents some other interesting properties of the modules constructed, one being that their simple quotients are holonomic.