Contrary to what the system text message say when the system first becomes available, they do carry over to the evening. What they don't carry over to is the next free day.
So, say, chapter 1 you have 2 points in the morning. If you use neither of them, you'll get an additional one in the evening for 3 in total. But if you go to bed without using them all, THEN they'll be lost forever. There's no real incentive to not use them during the day unless you really don't like any of the characters available, as in the evening of chapter 1's free day for instance there are only two events available at all (so you'd end up wasting one), but yeah, it is possible.
I wrote it right in the manual text and the help section of the notebook, but I made a misleading TL error that would make you think otherwise in that first explanation. Sorry for the confusion, that one's all on me.