Many bass instruments with primary resonators have disharmonicity affecting the tone: partial vibration modes that end up having higher frequency than a proper harmonic. Those blend better with higher pitches when those sharp overtones match the other pitches than when the fundamental is "correct".
Prime suspects here are string instruments with comparatively thick strings, like bass guitars. However, if the prime source of overtones is by distortion of the fundamental rather than by overamplification of physical overtones, disharmonicity should not come into play. So it's more a "funky" or "twangy" bass that will be affected than a "hard rock" bass sound.
This theme of "distortion of the fundamental" is also applicable to wind instruments operating on interrupted air streams, like reed and brass instruments (so they should now show disharmonicity) but not flutes. Flutes, however, tend to have rather few overtones and thus are not all that helpful for establishing a harmonic base.
The lower bass notes of a piano tend to require tuning lower than "appropriate" (stretch tuning), particularly of upright pianos which have shorter and thicker strings compared to grand pianos.
Bowed string instruments are rather complex since their overtones to a good degree come from the sticking action of the bow, but the bowing action itself causes pitch shifts in the attack phase as well as loudness dependent pitch shifts. This is partly compensated by today's bowed string instruments almost universally being fretless and thus amenable to on-the-fly corrections by the player.
Generally continuous-tone music instruments facilitate oscillators that are fed from some energy source. More energy is fed into the oscillators for louder notes, requiring more of a phase-lead and possibly resulting in a higher frequency for higher volumes. However, the tap-off of energy into sound can counterbalance this again, so the net result really depends on multiple balancing factors, and some aspects of musical instrument design may focus on making pitches reliable across different loudness by design or during the process of intonation.
TLDR: can of worms.