Hi nameless, it is normal that moving a polyline point will not snap to itself while dragging points but you can use the Transform > Move command if you need to do that.

If that's not it then it sounds like you have turned "Object Snap" off. It is controlled by this button in the bottom toolbar:



If that is not highlighted in orange click on it to turn it on.

It is also possible to disable specific types of snaps, that is done by clicking on the little arrow above the Object Snap button that will appear when you move the mouse over it. A menu will pop out where specific snap types can be disabled so you might see if you have unchecked any of those previously:



- Michael