Generally, yes, but it depends on the state. Every state has its own rules regarding nonresident returns.
For example, nonresidents with more than $33 in Pennsylvania-sourced income must file a return, while nonresidents with less than $600 in Missouri income don't have to file. If you're not sure, your best bet is to contact the Department of Revenue in that state, or visit their website. Most state websites have a section devoted to nonresidents and part-year residents.
Note: If the other state happens to be Alaska, Florida, Nevada, South Dakota, Texas, Washington, or Wyoming, you won't be able to file a nonresident return because those states don't collect income tax. However, you'll still need to report that income on your resident state return (assuming your resident state collects income tax) as well as your federal return.