I can only hear this effect on F6 or above, but if I use the sustain pedal, i can hear it from G5. The higher the note the easier it is to hear the low notes.
We call this sympathetic resonance and it happens when two strings are related via harmonics.
This is made especially obvious when we consider the harmonic series:
Begin by pushing down the G right above middle C without having it sound. Then put down the sustain pedal and play middle C; you may hear an upper pitch, which we'll talk about soon. Now try playing the C an octave below middle C while still keeping that G sustained; you should now hear that G loud and clear!
This is because that G is within the harmonic series of the second C that you played (the one below middle C) but not within the harmonic series of middle C. (Although, if you heard a higher pitch sounding when you played middle C, you actually heard a G above the one you were sustaining, because that higher G is within middle C's harmonic series.)
For a more advanced example, try silently holding down the B♭ below middle C. Then play the C below middle C, the C two octaves below middle C, and the C three octaves below middle C. The B♭ will be faint, but you should hear it when you play that lowest C.
If your piano is consistently sounding these lower pitches to a distracting degree, you may need to make sure that your dampers aren't faulty.