  In 1967 Harary, Hedetniemi, and Prins showed that every graph $G$ admits a
complete $t$-coloring for every $t$ with $\chi(G) \le t \le \psi(G)$, where
$\chi(G)$ denotes the chromatic number of $G$ and $\psi(G)$ denotes the
achromatic number of $G$ which is the maximum number $r$ for which $G$ admits a
complete $r$-coloring. Recently, Edwards and Rz\c a\.{z}ewski (2020) showed
that this result fails for hypergraphs by proving that for every integer $k$
with $k\ge 9$, there exists a $k$-uniform hypergraph $H$ with a complete
$\chi(H)$-coloring and a complete $\psi(H)$-coloring, but no complete
$t$-coloring for some $t$ with $\chi(H)< t<\psi(H)$. They also asked whether
there would exist such an example for $3$-uniform hypergraphs and posed another
problem to strengthen their result. In this paper, we generalize their result
to all cases $k$ with $k\ge 3$ and settle their problems by giving several
kinds of $3$-uniform hypergraphs. In particular, we disprove a recent
conjecture due to Matsumoto and the third author (2020) who suggested a special
family of $3$-uniform hypergraph to satisfy the desired interpolation property.
