There are examples all over the business of people working their way up through the ranks so yes it is definately possible.
My own observation is that the department to pass through is dispatch/turnround co-ordinator (I'm not talking airline operations people, although no reason they can't, but the ones on the ground). This gives such a strong grounding in the business and understanding of what can happen and how to resolve it.
That said I have seen people go from ramp agents to:
Shift Manager
Station Manager
Safety Manager
Training Manager
The same can be said of check in and dispatchers too.
GA