When talking about "bad input", you should consider where the input is coming from.
Is the input entered by a user or another external system you don't control, you should expect the input to be invalid, and always validate it. It's perfectly ok to throw a checked exception in this case. Your application should 'recover' from this exception by providing an error message to the user.
If the input originates from your own system, e.g. your database, or some other parts of your application, you should be able to rely on it to be valid (it should have been validated before it got there). In this case it's perfectly ok to throw an unchecked exception like an IllegalArgumentException, which should not be caught (in general you should never catch unchecked exceptions). It is a programmer's error that the invalid value got there in the first place ;) You need to fix it.