Yes, this can happen, but there is some randomness involved based on the team owner and the managers ahead of you. I played one league where I would manage in the minors, take over as GM for the draft (by going to commisioner mode) and then return to my old job. I got promoted from Rookie league up through AAA ball, but the owner liked the way the ML manager was winning with my talent so he kept getting 3 year extensions.
Maybe if I was more patient I would have gotten there eventually.
Be advised that managing in the minors can be a bit frustrating as the AI loves swapping players up and down and you seem to always have to be fiddling with your line-ups.