A map $f$ from a continuum $X$ onto a continuum $Y$ is $n$-confluent if each subcontinuum $K$ of $Y$ is the union of the images of $n$ or fewer components of $f^{-1}(K)$. Following on the work of T. Máckowiak [Dissertationes Math. (Rozprawy Mat.) {\bf 158} (1979), 95 pp.; MR0522934 (81a:54034)], V. C. Nall\ [Houston J. Math. {\bf 15} (1989), no.~3, 409--415; MR1032399 (91a:54019)] and Nall and Vought [Fund. Math. {\bf 139} (1991), no.~1, 1--7; MR1141288 (92h:54048)], it is proved that if $n\geq 3$ and $Y$ contains no $n$-od, then $f$ is $(2n-4)$-confluent. Examples are given to show that this result is best possible for all $n\geq 3$. In these examples, $X$ contains a pseudoarc, prompting the authors to inquire whether their result may be strengthened for $n>3$ if $X$ is required to be hereditarily decomposable.