Playing "on the one" might mean different things to different people. For instance, Bootsy was famous for playing "on the one" because James Brown told him to play on the one (or so my always suspect memory tells me), meaning that the emphasis for the bass line, in James Brown's band, was always the down beat. As Bootsy would say, "it all starts on the one, baby".
I take a different view to Bootsy and to some of the remarks posted above.
In my view, there can be compositions where the bass line, or bass instrument, doesn't have to be confined to providing a rhythmic underpinning (nor, for that matter, is there necessarily a need for drums). It seems to me, that "bass" can be said to be a frequency range, differing significantly in range from, say, a mandolin. But a bass instrument can be providing melodic content within that frequency range. And while it is doing so, a guitar, for instance, can provide rhythmic content.
(One interesting example that just came to mind is Jaco Pastorius playing with Joni Mitchell.)
And as for being "on the one", I think lots of interesting things can be played in a group setting where the bass line starts on the two, or the upbeat of the two; or where the guitar player is playing in four while the bass player is playing in five. There are seemingly endless possibilities for musical expression.
There need not necessarily be any hard and fast rule that says that the bass player has to always be on the one; unless you want to keep your job and the bandleader insists on it; like with Bootsy in James Brown's band.
