Depends on the establishment and the individual.
First you have to realize that to be promoted, there needs to be room for you
to work as a bartender - my place has LOW turnover for bartenders, so barbacks
don't get promoted often. The individual also has to be not only proficient
at barbacking, but also continually expressing an interest to management in
bartending.
I'd say if you get promoted in less than 6 months or so, consider yourself
lucky. If you're not a bartender within 2 years, and they've hired other
bartenders since then, start working on your resume.