Can I use nick name as a DBA?

Try talking to a trademark lawyer the next time, You are permitted to use any name you want provided it is not in use already in NYS. If you want more information,you might want to contact my partner James DeFelice whose avvo profiile will give you his contact information. He is familiar with company registration, which also requires a name search to see if the business name is already in use. (If it is you can be sued for using it even if it is your own name). It appears you were doing a borough registration and the better option is to do a state one.
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LEGAL DISCLAIMER
Mr. Sarno is licensed to practice law in NJ and NY. His response here is not legal advice and does not create an attorney/ client relationship. The response is in the form of legal education and is intended to provide general information about the matter in question. Many times the questioner may leave out details which would make the reply unsuitable. Mr. Sarno strongly advises the questioner to confer with an attorney in their own state to acquire more information about this issue.

Sounds to me as if the guy at the clerk's office wasn't a lawyer and was thinking about a different question. That question is: "MUST I register my name as a DBA if I use it as a proprietor of a business?" And in general, the answer is NO. If you are doing business under your own name, then the state government doesn't require a registration for what some states call a "fictitious name" or "business name."

But probably if someone wants to name a business "Mac&Lee" it would be necessary to register it. A separate question that would be good to discuss with a lawyer is whether to operate simply as a "general partnership" of two guys, Mac and Lee, or to establish a separate "entity" such as a corporation or a limited liability company ("LLC"). In the latter event, registration of the "entity" serves to register its name, automatically.

Check your state registry of corporate/business names. If it is available, the you can, otherwise you cannot.