Yes, it is possible to create your own name servers within your web hosting account. Many people choose to do this because they fell it gives them an added sense of privacy and/or professionalism when people conduct a DNS lookup. The WHOis database publicly displays the names of your name servers, which is a reliable indicator of which hosting company you use. If you would like to keep this information private, or display another name for your clients or site visitors to see, then you can install DNS software within your hosting account and create name servers using a domain that you own.
It is important to note that you cannot install DNS software within certain hosting plans, including shared hosting. Thus, if you'd like to create your own name servers for your domain you will need a dedicated or VPS hosting plan.