I mostly agree with Carl Witthoft's answer, though I'd say the procedure suggested there likely falls under what the question described as "unconventional methods." Basically, if a sound is created by one sounding body, and another body is present with a resonant frequency among its harmonics that matches the first sounding body, then you can get a special effect. For example, a string may sympathetically vibrate when one of its harmonics is sounding.
Note the first string doesn't respond sympathetically at its own fundamental (which might indicate the presence of true undertones). Instead, some rather small part of the vibrating energy will "leak" (due to the imperfect nature of real vibrating bodies) into the fundamental mode (as well as other modes/frequencies). If anything, this effect is further proof of the pervasive nature of the overtone series, which causes this all to happen. If the second body didn't have overtones at the right frequency, there would be no sympathetic vibration and no leakage to the fundamental.
This could easily be confirmed experimentally with some audio software capable of doing an FFT or even by damping lower vibrations on the second body to see what's really going on. For example, suppose you created a sound of 300 Hz and put it in the presence of a string with a fundamental frequency of 100 Hz. The string will resonate at 300 Hz sympathetically. But assuming the stimulus is strong enough, some energy will eventually leak into the other modes of the string, including 100 Hz. If this were truly evidence of undertones, you'd only see vibration at 100 Hz. However, a real-world string will begin to have some limited vibrations (due to similar leakage of energy) at other modes too, e.g., 200 Hz or 400 Hz, which have no relationship to the first string's frequency in terms of overtone or undertone series. (And no, you can't blame 200 Hz as sympathetically vibrating with the 600 Hz harmonic of the first sound, particularly if a 300 Hz sine wave is your stimulus.)
In sum, a true "undertone series" like the overtone series is not produced naturally by normal harmonic sounding bodies. Under certain circumstances, some sounding bodies can be manipulated to vibrate at unusual modes (including undertones), and other sympathetic bodies in the area could leak energy to other modes of vibration.
From the question:
I can sometimes hear a note which is an octave lower than what's actually being played on a piano. This can be very significant if this note is in a chord, which can even make me think it's an inversion.
There are a variety of effects that can create that sensation. Without knowing the exact circumstances, I can't say for certain, but my guess is that what you're hearing in a situation like that is the psychoacoustical phenomenon known as a missing fundamental.
This is a well-known cognitive effect where the brain creates the sensation of a fundamental that matches the harmonics in a stimulus even if that fundamental frequency isn't present in the real sound. The reason for this is presumably because the brain spends so much time processing harmonic stimuli and condensing all of those various overtone frequencies into a single fundamental "pitch" that we hear. So, when we hear a bunch of frequencies and no fundamental, the brain inserts it anyway. (Note that this effect is exploited all the time, e.g., in telephone speakers which are often incapable of producing the fundamental pitches of low male voices. Instead, the speaker transmits the higher partials, and your brain adds in the fundamental so you perceive the voice at the correct pitch.)
Anyhow, by listening carefully and playing notes of, say, a major triad, you can sometimes have a sensation of hearing the fundamental that would theoretically lie under those pitches, generally an octave or two lower (depending on chord voicing). This effect becomes much more pronounced when playing the triad in just intonation. And if played with pure tones (sine waves) in a 4:5:6 frequency ratio, many people would simply hear a fundamental pitch two octaves lower and no individual frequencies at all. (Piano tuners and other musicians who train themselves to listen to harmonics may not hear this "ghost" effect sometimes, however.)
I should note, in closing, that psychoacoustical phenomena such as the "missing fundamental" effect are likely one of the primary reasons why an "undertone series" was first postulated. There are many situations where the brain can seem to "hear" lower pitches than are actually present in a stimulus. Nowadays, with access to audio analysis software, it's easy to verify which frequencies are and aren't present in a given stimulus.
