Wow. So many different answers.
YES! Absolutely.
Once you purchase/register a domain name, that name is registered under your name for the period you registered it for, and you cannot change the name that is registered under the Top Level Domain (extension, such as .com, .net or .org). This is also true if you used a Second Level Domain (such as .co.uk, .me.uk, .org.uk) because that name is reserved for the period of registration under your name and contact info.
However, I don't believe that is what is being asked. That seems to be confusing. You can always nix the old domain and register a new one to use with your account.
Depending upon your web hosting account, you can probably add a domain name to the account, too (use both domains to preserve old traffic and just market the new one), but you can get into issues with duplicate content, so it is best to scrub the web of the old links.
Starter web hosting accounts will usually require an upgrade to add another supported, hosted domain (or a few).
Places that give out free domains will likely allow you to upgrade to a paid account.
But under any circumstance, you can always move to a new hosting provider.
But yes, you can always register a new domain to use with your website. If your website host account doesn't support multiple domains, you can always reconfigure the account to use the newly registered domain name that you would prefer to use.
Also... you can change the registration in the case you sell a domain name, too. that is a different topic, but allows you to sell a domain name registration before the registration expires.