Does this kind of survey make any sense? Why should somebody answer honestly if the survey is not anonymous and you don't want to admit something? If you want to admit something, such an survey may at best catalyze it. Therefore I would expect that the result of such an survey just does not reflect reality.
You could cross check answers of a well designed survey to get an indication if the answers are honest or not but this does not nearly bring the result close to an anonymous but more honestly answered survey.
Exactly. So many comments here assume that students wouldn't lie on this survey, which I think is an ridiculous assumption. From my experience, students are afraid to be overly honest on anonymous feedback surveys just because of the possibility teacher will recognize the handwriting. There's no way they would fill in truthfully a non-anonymous survey that could get them in a world of trouble.
You could cross check answers of a well designed survey to get an indication if the answers are honest or not but this does not nearly bring the result close to an anonymous but more honestly answered survey.