Can I get my citizenship if I have a 3rd degree assault that has been dismissed over 15 years ago?

Usually a criminal or traffic charge that has been dismissed will not stand in the way of eligibility for naturalization, but a major exception can exist where the USCIS concludes that a dismissal took place following an acknowledgement of the elements of the offense. For example, a dismissal following a pre-trial diversion where the defendant has aliquoted will be treated as a "conviction" by the USCIS. In the naturalization application process it will be necessary to disclose all arrests and citations and to provide a full set of court-certified disposition documents (or court-provided explanations about why disposition documents no longer exist). For this reason you will need those documents both to determine naturalization eligibility and also to support an application for naturalization. There really is no substitute for consulting with an immigration attorney who, after reviewing the records and learning all of the relevant facts, would be able to advise you about eligibility and could offer legal representation in the application process - a process made significantly more complex because of a record of arrests/citations.