Depends. In many cases of antibiotic resistance, the actual "resistance" is a matter of degrees. For example, Penicillin interrupts the cell walls of bacteria, so in Gram negative bacteria it is only active in the periplasm. Some forms of Penicillin resistance work by metabolizing the Penicillin before it can do its thing. Other forms work by simply pumping the Penicillin out of the periplasm. The former mechanism requires a specific enzymatic activity. The later usually just requires tuning the pre-existing pumps that maintain the environment of the periplasm to be more specific to Penicillin or pump faster over all. In fact, most "multidrug resistance" genes are simply pumps that can exclude drugs from the insides of bacteria in general.

But really, this is just the shortest of short answers. If you want to know more about how these things can evolve, you should read about Richard Lenski's long-term experiments: http://en.wikipedia.org/wiki/E._coli_long-term_evolution_exp...