**Abstract: **
We consider the Hill operator
$$ Ly = - y^{\prime \prime} + v(x)y, \quad 0 \leq x \leq \pi, $$
subject to periodic or antiperiodic boundary conditions, with
potentials $v$ which are trigonometric polynomials with nonzero
coefficients, of the form (i) $ ae^{-2ix} +be^{2ix}; $ (ii) $
ae^{-2ix} +Be^{4ix}; $ (iii) $ ae^{-2ix} +Ae^{-4ix} + be^{2ix}
+Be^{4ix}. $ Then the system of eigenfunctions and (at most finitely
many) associated functions is complete but it is not a basis in $L^2
([0,\pi], \mathbb{C})$ if $|a| \neq |b| $ in the case (i), if $|A|
\neq |B| $ and neither $-b^2/4B$ nor $-a^2/4A$ is an integer square
in the case (iii), and it is never a basis in the case (ii) subject
to periodic boundary conditions. This is a common work with Plamen Djakov.