In what case does a Coil with Current flowing through it NOT produce a Magnetic Field? According to what I've studied, when Current flows through a Coil, a Magnetic field is produced.
Recently, I've been told that there is a case in which Current flows through a Coil Without producing a Magnetic field.
I've searched everywhere, but I haven't been able to find this situation.
So; What is the case in which current flows through a Coil without producing a Magnetic Field?
PS: Please Strike out the idea that the case doesn't exist; the one that told me of it is a Physics Professor with a wide array of knowledge. I lost contact with him though; otherwise I would've just asked him...
 A: It's called a Bifilar or Caduceus Coil Which contains two closely spaced, parallel windings. Most often used for constructing wire wound resistors in order to minimize both inductance and stray magnetic fields
A: The person you've been talking too is probably imagining a physical case in which each clockwise coil is matched with a counter-clockwise one. This is probably easiest to realize by wrapping once along a solenoid one was and then wrapping an over-layer in the reversed sense.
If the matching is good and the coil density is high and uniform the resulting field will be net zero at all places farther from the wires than the spacing between the corresponding wraps.

Arguably each stretch of the coil does produce a field, but they cancel each other out almost everywhere.
A: I think the only time would be when the coil was wound one way and then the opposite was with the same number of turns so that the net magnetic field produced was zero. In fact the coils would have to be wound particularly carefully and the cancelling would probably not be perfect everywhere.
A: There is another case besides the one mentioned in other answers: In inductors and transformers, an internal magnetic field is created, but there is a closed magnetic loop which means that the magnetic field does not escape. Besides avoiding creating interference, it is also desirable to prevent escape of the magnetic field for efficiency reasons. The best type of transformer in this respect is the toroidal transformer, due to its circular symmetry.
https://en.wikipedia.org/wiki/Transformer
https://en.wikipedia.org/wiki/Inductor
As others have mentioned, this construction is not suitable for wirewound resistors because of the inductance. Wirewound resistors frequently have coils wound in opposite directions to ensure a pure resistance without any undesired inductance.
Another device worth mentioning is the "humbucking" type of guitar pickup. This is a design with two oppositely wound coils close together, which avoids picking up interference from the ambient magnetic field. It's not normal to pass a current through such a pickup, but if you did the magnetic field produced would be present but extremely localised (due to the small spacing between the two coils.) It's essential that there some localised interaction to be able to pick up the magnetic field of the string. The closer the coils are together, the smaller the area of influence. It's not possible to have a guitar pickup in which one coil is co-wound with the other as then it would not interact with the string at all (and would produce no field when a current was passed through it.)
https://en.wikipedia.org/wiki/Humbucker
A: The only thing i can imagine is a coil constituted by one wire which one half is coiled clockwise and the other half is coiled counterclockwise. So one part produces a magnetic field in one direction and the other part produces the field in the opposite direction. 
This might be obvious, but I really cannot think in any other possibility for this, unless the wire is not in any normal condition (e.G.: extreme temperatures, or other extreme conditions involved).
