There are many different mechanisms for antibiotic resistance and consequently many different answers to the general question here.
Let's start with the lab case of a bacteria specifically transformed with a plasmid which contains a gene encoding antibiotic resistance as a marker for maintaining the plasmid. Note here that we are talking about an engineered plasmid construct, and the antibiotic resistance is used for selective pressure against "losing" plasmid. By itself, the expression of betalactamase or another antibiotic resistance gene does not incur a significant growth penalty (it does a little and in a very competitive environment it may be somewhat significant, but not so much in a rich media). What does give a significant penalty is the rest of the plasmid, for various reasons that will take a long time to go through. Importantly, for most plasmids, there is nothing perfectly dividing the plasmids between daughter cells during division (unlike the genome) and so some cells will naturally after division have more or fewer plasmids. If there is no selection pressure to maintain the plasmid, eventually the cells with fewer or no plasmids will outcompete the ones that are copying dozens of copies of the plasmid. However, this process will hardly ever completely remove plasmids from the total population. As anyone who has cured out plasmids from E. coli will tell you, it can take multiple subcultures (100s of generations) and even then you need to check individual clones to make sure the plasmid is completely gone.
Multiple-resistant bacteria in the "wild" do not gain resistance to all of the different antibiotics by capturing lots of plasmids though. Most of the MR bacteria have general mechanisms for resistance (such as efflux pumps that recognize multiple different antibiotics) as well as a few specific resistance genes that are either a part of the genome or are contained on plasmids with other essential genes such as they will not be removed by simple dilution. However, the general mechanisms of resistance do tend to be quite deleterious either in direct energy expenditure or in requiring suboptimal choices with regard to doubling time.
However, once again, its important to remember that resistance is a phenotype of the population. In bacterial populations it's quite common to only have some subpopulations expressing certain genes at certain times. A populat mechanism for efflux pump expression for example is to have a kind of oscillating gene expression circuit. At any given time a cell will occasionally express the gene, but often will not. That way a given cell can by chance happen to be resistant at a certain time when the antibiotic is introduced. The cell thus pays only a fraction of the cost of maintaining resistance at the cost of only being resistant for a fraction of the time. If the oscillations are not synced in the population, some subpopulation of the cells will be resistant at all times. When an antibiotic is introduced, the expression is up-regulated and those cells that were resistant survive and reproduce, thus the overall population is saved. This is just one mechanism, but should give you an idea of how the bacteria can get around always maintaining resistance genes.