Typically sympathetic resonance generates more copies of the same harmonics or fundamental tones, not new tones in between as your post suggests. I can think of one or two phenomenon that would describe this but I'd surprised if you can hear these if the notes are not close together (e.g. maj or min 2nd).
When two harmonic waves (i.e. notes) are played together they will combine to form an interference pattern (both in space and time). In the time domain the 2 tones combine to create a sum tone which is the average value of the two fundamental tones and the envelop of this oscillates at a rate determined by the difference tone. When the notes are really close most listeners will hear the average pitch (not the two fundamentals) and perceive a volume fluctuation or beating of the sound (in and out like someone is using a volume pedal). Guitarists use this to tune strings, listening for the beats to disappear.
The phenomenon will occur in theory for any two tones but for you to hear the average I think the separation of the fundamentals needs to be withing a critical difference for pitch discrimination (that's why I'd be surprised if you are really hearing this with the fundamentals most people hear either 2 notes or their average).
All harmonics is the series can also generate these beats and averages but I'd expect them to be difficult to hear as independent notes.
Another thing that could be happening is an illusion generated by aural harmonics created in the ear. This is a well known phenomenon but usually does not generate average tones (as far as I recall).
For it to be sympathetic resonance there would have to be some copy of this in between note in the original spectrum and something in the instrument tuned to that difference. This is highly unlikely. Also, in linear systems you cannot excite the average tone. This is an artifact of wave superposition and linear systems do the opposite, they decompose superpositions into pure tone components. A linear system will not detect sum and difference, it would detect the original 2 tones and vibrate to those frequencies. The ear is non-linear and there are limits to perception in the brain leading to these kinds of aural illusions.
You could help make the question easier to address by giving some specific examples, actual notes on the piano and what you think the 3rd in between note is. Also, if you have access to a mic and a digital signal processor (data acquisition card or fft software like Transcribe) you could see for yourself if the note is really there. If not then I'd say that's evidence that it was an aural illusion, if so then some mechanical coupling in the instrument is at play and that would be interesting to discuss.
In short I'd recommend reading up on sum and difference tones, and aural harmonics.