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.