Yes, you should never be double-taxed on the same income by two states.
Think about where you lived when you earned the income. If you lived in Massachusetts, then moved to Connecticut, then moved to New York, you may need to file a Part-Year Resident return for all three states, reporting the income you earned while living in each those states.
Some states ask you to exclude income; others take all income and apply a % based on time lived in that state, so don't be discouraged if the interview doesn't directly ask you to exclude income. Do your MA return first, then CT, then NY.
If your W-2's don't break this down properly for you, you may need to do the calculations yourself. In the MY INFO section, be sure to indicate that you 'earned income in other states' (MA, CT) if your address is a New York address.
Here's more info on How to File a Part Year Resident Return.
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