When you know your insurance will expire in a month (e.g. loss of student status), you can't go to arrange your health insurance as early as mid-September, you have to wait until the end of the month. This is stupid.
Besides, you could automatically get a bill home as soon as it's obvious you're not insured anywhere... and the Health Insurance Fund could automatically obtain from universities/high schools information on who has student status, without having to go around carrying certificates.