Can unwanted coupling drive signal lower than intended?
This is just to address your initial bad assumption, but maybe not answer your whole question or solve your real problem.


  I do understand that two conductors can become inductively coupled, and that current change in one conductor induces voltage in other conductor. By looking at the oscilloscope trace, I see that signal voltage drops- my assumption was, that only more positive voltage can be induced.


As you suspected this is a wrong assumption.

If two conductors are inductively coupled and a (change in) current in one direction of conductor "A" induces a positive voltage in conductor "B", then a (change in) current in "A" in the opposite direction is going to induce a negative voltage in conductor "B".

In any case where at some instant in time you see a positive induced voltage in some conductor, you're likely at some other time to see negative induced voltages, just because currents don't tend to increase or decrease monotonically toward infinity.

Note: This does not mean I think that the problem you are seeing has anything to do with inductive coupling.