  The envelope theory is an easy-to-use approximation method to obtain
eigensolutions for some quantum many-body systems, in particular in the domain
of hadronic physics. Even if the solutions are reliable and an improvement
procedure exists, the method can lack accuracy for some systems. In a previous
work, two hypotheses were proposed to explain the low precision: the presence
of a divergence in the potential or the lack of a variational character for
peculiar interactions. In the present work, different systems are studied to
test these hypotheses. These tests show that the presence of a divergence does
indeed cause less accurate results, while the lack of a variational character
reduces the impact of the improvement procedure.
