Yes, this happens from time to time. It depends on the laws of the state. It is more common for a city to annex land, such as from an unincorporated county. But cities do divide and become their own jurisdictions too. Why they do this depends. Sometimes it can be because the city grew too large, or because of a difference of opinion.
The answer to this is going to be different depending on the state in which you live. Governments below the federal level are governed by state law. State law determines when or if a community can detach itself from a higher level of government. Perhaps if you could tell us where you live we could be of more help.
so how would we go about getting this done in bristol township, bucks county Pa, the reason for this is simple, the council and school board are taxing us to death, we have our population is just under 10,000, an ample amount of residential properties and a good amount od commercial business within Croydon, also a national park and the Delaware river and a potential for much more growth