Generally you should but depending on the type of test it might be prohibitive. I've definitely written up tests for validation purposes that I wouldn't use in a test suite normally before. Said tests usually require some complex or brittle setup and take hours or days to run. Because of that said tests are kept around for rare occasions.
Because they take so long and/or are so brittle, it's easier to just keep them on the side and update them when needed rather than deal with the back and forth on that in a project with a lot of code churn (i.e. when updating or retrofitting a legacy project). Once again, it's not ideal but it's better than nothing.
