How to conclude the nature of interference for such scenarios?
In some cases, \$V_{cm}\$ is an AC voltage produced by current in a  ground loop (in orange, below).
It cannot be shorted out.
If this ground loop can be broken, no current can flow around the loop, so no voltage is developed across Rw.
One way out: disconnect (float) \$V_{sig}\$ so that Y doesn't connect to ground. 

In some cases, the signal source cannot be floated, and is unavoidably grounded. And it is likely that the DAQ is grounded as well. So the loop cannot be easily broken.

Adding extra ground wires likely increases loop area or loop current. \$V_{cm}\$ gets worse, or at least doesn't decrease. A very annoying problem to solve, especially in complex measurement systems where lots of equipment is grounded.