It depends on the states involved.
Since you stated that you live in TN (which has no state income taxes on earned income) but worked in KY (which has a state income tax on earned income), you will need to file a KY nonresident state income tax return to report your KY-sourced income.
Usually you will get a state income tax credit in your home state for state taxes paid on income taxed in another state. But since TN have no state income taxes on earned income, you will not be able to claim any state tax credit in TN.
Here is additional
information about filing in multiple states (select "see more answer"
to view the entire attachment)
https://ttlc.intuit.com/replies/3300797
