Are there any special provisions provided for students who have schizophrenia (for example)? Is there an "adjustment" to grading? Or perhaps there is no adjustment to the grading process per se, but only to the due dates? Is there anyone here who has been in this type of situation?
Every university I have been affiliated with (in the U.S.) has had an office of disability services, tasked with evaluating individual cases and making (binding) recommendations to instructors.
If your university has such an office, then you should refer this case to them and avoid judgement calls of the type you ask about.