We hear this so often. We have had tenant's show up with a U-Haul and all their worldly belongings inside. There are some legitimate reasons that a tenant may be in this position, but consider this . . .
Notice's to Quit are for three days, once a tenant receives a Notice to Quit, they are in a position that they need to move right away. If they show up with their belongings with them - or - if they tell you they need to move right away, listen to their story, but be very careful - the tenant in front of you may be in the middle of an eviction.
Legitimate tenant's give prior landlords 30 days notice. If they aren't planning on giving their old landlord 30 days notice, don't expect to get 30 days notice when they move out of your place.