Timbre describes the spectrum of a sound, most noticeably, the distribution and relative strength of it's overtones. Most acoustic instruments are approximations of a one-dimensional oscillating body, usually either a string or a column of air. Such a body will intrinsically vibrate with overtones that align with the harmonic series, leading to a harmonic timbre. 12-TET is a compromise system that attempts to approximate the consonances of such a harmonic series (the 2nd, 3rd, and 5th overtone being equivalent to the octave, perfect fifth, and major third), while also having the property of isomorphism (all the intervals are equally-spaced).
However, there are two cases of acoustic instruments that I can think of where its timbre is somewhat out of line with a harmonic series, a property called inharmonicity.
The first case is instruments that do not approximate a one-dimensional object as closely. These are typically pitched percussion instruments, especially bells, and to a lesser extent, marimbas, xylophones, tubular bells, etc. In this case, you have a 3-D body vibrating in multiple dimensions, with considerably more complex overtones. This is what leads to these instruments generally having more of a characteristic "clangy" sound. And even still, these instruments are generally carefully tuned to and played in something like 12-TET. At the other extreme, if such instruments are not tuned, you can get a massive wave of sound, such as with a gong or cymbal.
The other case is pianos, where the thickness of the string (relative to length) combined with its rigidity, cause the higher overtones to be slightly sharper than the ideal harmonic. Piano tuners have to take this subtle effect into account; in order to make the piano sound "in tune" with itself, the higher keys have to actually be tuned slightly higher, and the lower keys slightly lower relative to the same note in a more central octave. This technique is called stretch tuning. The inharmonicity that produces this effect is also more pronounced in smaller pianos (where the strings have to be thicker) which is why they generally have a less-pleasant, more "clangy" sound.
Here's a video about inharmonicity, in the context of a bell.
And here's a video about stretch tuning on a piano, which also points out that guitar makers have to take this effect into account when placing frets.