Yes, you can run a wire separate back to the main panel for retrofiting grounded receptacles in an ungrounded old installation.
The grounds and the neutrals are the same at the main disconnect point (usually your main pane).

Also, if you look at one of the boxes, where the wiring enters the box, you may find the ground wire wrapped around the sheathing and bare to the clamp. The clamp is tightened and thus the box grounded. A receptacle or switch screwed into the box would also be grounded. I'm not saying it was a good way of doing it but that was one way they did it.

The ground wire is supposed to follow the other circuit conductors, even if it was added later.

This is the reason non one usually does what you have described (and it does not sound like it is installed correctly).

The added ground conductors are not allowed to take any random path.

Ok just to confirm, the bare copper wire goes up and joins the other random copper wire strands and feeds back to the box. It literally spider webs. They are not individually routed to the box they connect to one another.

So I could safely run a bare copper wire from the spiderweb down the wall and connect it directly to the outlet (not outlet box but the actual grounding spot on the outlet)?

Thanks

" random copper wire strands "

To clarify, there should be no random strands if the grounding was done properly. For each circuit, you should have 1 ground. It should follow alongside the other wires in the circuit and it should connect all the way back to the panel. It should be done in a neat and workmanlike fashion and should not look like a spiderweb or any other mess. As a general rule, if you look at any electrical work and your first reaction is "What the heck is this mess!", then it probably wasn't done properly.

From your description, it doesn't sound like the retrofit was done properly. It might be "better than nothing," but not by much. The ground is a safety device that should provide an uninterrupted path for any stray current to flow immediately away from you and anything else in the house.

Bottom line - if you can access these areas from above, then you should just run all new cable instead of a just a ground. If you are fishing wires, you may as well just do it once and know it is done properly.

I would love to replace the wires, but I do not want to gut all the drywall that my wife just painted, and I know there is a reason for it but I hate the fact that people staple the wires, otherwise I could of easily pulled out old while pulling in new.

The copper wire above is not disgusting it is just that it goes up and then connects to other copper wires which then go further towards the box and they meet up with others along the way. So many outlets are actually joined together via the grounding wire even though they are not on the same circuit. Is this OK?

No, it isn't OK.

When you run new wires, you do not have to remove all the old wires. You can just disconnect them and leave them in the walls.

Sorry, brick not true. ARt 250.130(3) covers retrofitting or extending grounded receptacles on a non-grounded branch circuit. There's no requirement that the grounding conductor follow the current carrying wires. It does have to go to one of the listed items of which "a spider web of bare wires" may not meet. However if the central core of that mess is part of the bonding of the building grounding system it might be OK. Running direct back to the panel (even if the branch circuit wiring doesn't go that way) is also allowed.

Any practical way to test the ground to see if it is sufficient? I am assuming the little outlet tester will not be adequate.

No, there is no way of "testing." What you need to do is run the ground to one of the known good points. This could be the main panel or one of the points in the grounding system (typically ground rods and the incoming house water service) or the stuff that bonds them together.

By any chance is this house located in... Maine?

"Pharkus!... paging Mr. Pharkus... we have an ee-lek-trik-al issue on aisle 3..." :D

And 250.130(C) does not allow a " spider web of bare copper wire" to create a grounding path

The wire has to run back to the GEC or the panel.

They may not be run to each other and then back to the panel.

For correct operation of the circuit OCD the ground need to be in close proximity to the current carrying conductors.

If a large loop area is created between the grounding conductor and the current carrying conductors OCD operation will be slowed by the inductance introduced in the circuit.