The three personalities come from a PARALLEL (or howerver you spell it) UNIVERSE!
I'm serious.
Basicly, in a parallel universe, Rock never existed, and Wily reprogrammed 6 of Light's robots. Tima Man and Oil Man were not, and thus they volenteered to be converted to fighting robots, and they fought Wily ever since. Along the way, Time Man meets Splash Woman (I'm thinking, between the 2nd and 3rd game), and Splash Woman joins in on the group.
Note: In this universe, Time Man was actually created by Wily, but because he couldn't get the time-control mechanism to work at all, he abandoned him. Light then found Time Man, who then worked as an assistant.
Yes, indeed, There is a "Time Man X Splash Woman" thing.