When bring 2 points together (right-click > Average) and then I try to join them together (right-click > Join), I receive an error message saying: "the selected objects cannot be joined as they are invalid objects":
Are the points on the same path?
If it's a closed path, in the pathfinder panel click on unite. This should help in a lot of cases.
And/or try Object > Path > Simplify
The "Unite" thing kind of works. Although my only concern is that it may do other things than just merge the 2 anchor points.
How about just deleting (the topmost) one of them with the The Delete Anchor Point tool (-)?
It will change the shape of the layer.
In some (simple) cases, you can get away fine with bringing 2 points together, but in many cases it can make a mess, sometimes obvious from the beginning. It may be better to make changes in other ways, also to fully obtain the desired shape.
A screenshot could be useful; it is easiest for helpers if you use the Insert Photos button at the top of the reply box.
Or click once with the Delete Anchor Points tool (-).
Or as Jacob said 41 seconds earlier.
Ton, as you may remember, quite recently Susan was 45 seconds ahead of me.
It should be forbidden...
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Instead of doing it in two steps have you tried the shortcut Cmd(Ctrl)-Shift-Opt(Alt)-J? This will make a single point of two selected points.
Seems that if 2 points are part of the same closed vector shape, you cannot joine them, the solution is to delete the line between them, then before you join them together.