If you didn't live in the other state at any time in 2016, you will not have to file a tax return in the state where your bank account is. Money deposited in a bank account in another state is not considered income from that state.

However, it's not clear when you moved. If you lived in the other state for part of 2016, you will have to file part-year resident returns for both the old state and your new state (assuming both states have state income tax).