  We study the necessity of bump conditions for the boundedness of the
Hardy-Littlewood maximal operator $M$ from $L^p(v)$ into $L^p(w)$, where
$1<p<\infty$. The conditions in question are obtained by replacing the average
of $\sigma=v^{-\frac{1}{p-1}}$ in the Muckenhoupt $A_p$-condition by an average
with respect to certain Banach function space, and are known to be sufficient
for the two-weighted maximal inequality. We show that these conditions are in
general not necessary for the boundedness of $M$ from $L^p(v)$ into $L^p(w)$.
