How many excuses are too many?
Whatever general advice this forum can offer is in my opinion extremely unlikely to offer much insight into your specific situation (though I'd love to be proved wrong by other answers). As you said, this is an issue of departmental policy, so you need to have a good long conversation about the situation with your department chair or grad/undergrad (as applicable) program chair. It is their job to set a policy on such matters that is legal and reasonable and provide you with any resources you need to do your job effectively and without creating an unreasonable workload for you (source: I am a department chair). And of course you will be able to provide them the necessary details that you cannot mention here. Good luck!