It depends on which state is your resident state. Your resident state will tax you on your income worldwide, no matter where earned. But a non-resident state will only tax you on the income earned in that state. If you were a part-year resident of each state during the year, then each state should only tax you on the income earned while you were living in the state.
Therefore, if you live in Massachusetts but worked in Vermont, then, yes, the Massachusetts return should include the Vermont income, but then the Massachusetts return should list a credif for the tax due to another state. In this scenario, file a non-resident return in Vermont and then create the Massachusetts return so that it will pick up the tax paid to Vermont.
If this is a scenario where you were part-year residents in both states, then, no, Massachusetts should not tax you on income that you earned while living in Vermont. You would need to file part-year resident returns for both states and allocate the income between the two.
Below is information on filing a non-resident return in TurboTax:
Plus information on filing a part-year resident return and allocating income: