The Availability setting determines when the student can submit work:
Do not confuse this with the Allow access settings under Restrict access. The Allow access settings determine whether the student can see the assignment on the course page, or whether the assignment is completely hidden from the student. After the student sees and selects the assignment, the Availability setting determines whether the student can submit work to the teacher.
Note that there is a Due date and a Cut-off date. If you do not enable a Cut-off date, the assignment will stop accepting work on the Due date. If you enable and enter a Cut-off date, the ...