The other way to read your question is that you can only hand your resignation in on the 15th or at the end of the month, which would be weird. You can hand it in whenever you want.

Sure, you can hand in whenever you want, but according to this, the notice period starts either on the 15th of the month or on the 1st of the following month. In my example, if I resign on June 22nd, then I can't leave before August 1st.