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Recently one wall of outlets quit working. Testing shows the hots are all good and the grounds are all good but the neutrals are all dead. They are all connected to each other but somewhere have lost the connection to the panel.

Can I add a neutral from another nearby outlet which will make these outlets work without tearing up the walls to find where the neutral has become disconnected?

No, you cannot do this. Assuming the nearby outlet you want to borrow neutral from is on a different circuit, you will be overloading its neutral. Even if it is the same circuit, it's very bad practice to leave broken wiring energized without fixing it -- what if it moves a little and reconnects just enough to arc? Then your house is on fire.

However, fixing this properly probably isn't as much work as you think. You most likely will not need to tear up your walls. Unless you recently drove a nail through a wire, the wire inside the walls usually doesn't just break. Rather, things usually go wrong at the outlets themselves. The so-called "backstab" outlets are especially prone to having the wires come loose over time. So the fix for this will involve pulling out each outlet in turn, making sure all screws, wirenuts, etc are tight, and ideally replacing any backstabs in the circuit. Most likely you'll find one of the outlets had the white wire come loose, and when you tighten it, everything starts working again.

Absolutely not. That sort of thinking works with safety ground, which does not ever carry current, except during a fault condition (and we hope there aren't 2 independent fault conditions occurring at once).

However, current flows in loops. Neutral is the normal "return half" of the loop, and it needs to be dedicated to this circuit. In fact, neutrals don't have fuses and rely on the circuit breaker on the hot side to protect the neutral. You can't have two circuits sharing a neutral, or you get this!

But there's another problem. This is AC power. It reverses polarity 100/120 times a second, which throws electro-magnetic force into everything around the wires. That's why transformers work. To keep that from causing a lot of problems, we require that all the current that goes out one wire comes back on another in the same cable or conduit. Among the wires, currents are equal and opposite, so their electro-magnetic forces cancel each other out.

Now, borrowing a neutral means current is now making a big loop - "out" (really, it's AC) this circuit's "hot" wire and returning via another circuit's "neutral" wire. That will throw EMF on both circuits, causing all sorts of mischief - eddy current heating, wire vibration leading to noise and metal fatigue leading to cracking and spot heating, etc. Nope.

Because of this, all practical wiring is done on a physical "tree" topology -- a branch may "split", but it may never "loop back" onto itself or another branch (except in one particular way, and even then, only in the UK where this is officially sanctioned).

Now, it can be hard to see when you're not practiced in a field. But this is most likely an easy problem. Wire breaks inside walls are rare. Wire problems are almost always at terminations of the cable run... And by definition, this must occur at either the last working outlet or the first failed outlet in the sequence.

As Nate discusses in his answer, backstab connections are the usual culprit. You're best changing it to a screw connection, but current Code advises using an $80 torque screwdriver, because "too loose" is more of a problem than "too tight".

If it's at the service panel, the neutral may go to a neutral bar, or a GFCI or AFCI breaker. Torque matters here too.

Lastly, if this diagnosis is given you by a 3-light tester, know that some of us call it a "Magic 8-ball" tester. It is made as a pass-fail device for new wiring e.g. During new construction. When troubleshooting old existing work, the legends can be wrong to the point of whimsical. The lights are useful; they save you the trouble of making 3 measurements with a neon tester.