Can I trademark a name that is already being used by a business that had not trademarked it?
~ New Haven, Connecticut
Maybe. Here are two important ideas you are missing: First, trademark common law rights happen automatically. Second, trademarks are really about preventing confused consumers.
Trademark Common Law Rights
Whenever commercial activity occurs, anything that consumers used to identify the source of the commercial activity gains a tiny bit of trademark protection. The more commercial activity, the more protection. This is called trademark common law rights.
Your question does not mention if the business name is merely used as an identifier (like the name of a trust or holding company), or, if it is used to engage consumers in commercial activity. If there is commercial activity going on, there could be trademark common law rights.
Trademarks protect against consumer confusion for the particular commercial activity. So, if the commercial activities used in connection with your trademark are not similar to the commercial activities of the other business, then perhaps there will be no confusion and the trademarks can co-exist.
On the other hand, if the commercial activity is similar or related, then likely consumers will be confused, and, infringement may occur. In trademark law, it is not necessary to prove that consumers are actually confused (this can be difficult). It is only necessary to prove that consumers are likely to be confused.
To understand how common law trademarks work in your state, you might want to consult the Model Uniform Deceptive Trade Practices Act. Also, the INTA has the text of the Model State Trademark Bill. At this time, 46 states have passed some version of this bill.
Finally, you should know: the nuance of this question is way tricker than you might imagine. There are 50 states, and, all states treat trademark common law rights slightly differently. For a question like this, you should find a local trademark lawyer that can help. Likely, the consultation will be short and to-the-point.