While the linked video is explained as an amateur (high school) orchestra in which the musicians haven't fully developed their intonation, there are situations in which classical musicians play deliberately out of tune. (Of course, since they typically do this at the composer's instruction, one could say they really aren't out of tune at all.)
The prime examples of "out-of-tuneness" would be those in the realm of microtonal music. Microtonality broadly refers to contemporary music (and sometimes music from non-Western cultures) that is composed outside the accustomed equal-tempered scale.
One example: Krzysztof Penderecki's Threnody for the Victims of Hiroshima. In this piece, the instruments are, by intention, out of tune with each other.
Detuning to achieve a specific pitch
Traditionally, the lowest pitch on a string bass is E, but there are scores that require an Eb. To do this, the bass player retunes the E string to Eb. Technically, the string is now "out of tune", but also plays an accurately tuned Eb.
See How do you play a low Eb on a double bass?
Sometimes a composer will ask a string player to "detune" their instrument for some passage of music.
For more on this, plus some examples of other "detuned" instrument and pieces of music that employ the technique, see Are there pieces that require retuning an instrument mid-performance?
Some composers or ensembles work with toy instruments which are (more or less) inherently out of tune. One example, John Cage's Suite for Toy Piano.
"Correcting" for equal temperament
The compromise that equal temperament makes (12-TET, specifically) is that most intervals are "out of tune" in comparison to their more sonorous "just" relationships. When I sang in unaccompanied choirs, the voice with the chordal third would often adjust their pitch up or down, depending on the needs of the chord. Similarly with fifths. String, brass, and woodwind ensembles will do this as well. (Basically, as long as there's no piano. ;-))
One post that discusses this is Making part of a triad chord sharper, but is it counted as minor?
There is a strong case to be made that the only "in tune" scenarios are those adhering to "just" intonation, meaning that intervals are integer ratios of frequencies. A "temperament" is a tuning in which those ratios are adjusted — "tempered" — to achieve some musical purpose. For example, 12-tone equal temperament — the adjustment of intervals such that 12 equal-sized semitones divide each octave — allows for the practical use of 12 different keys, but only by making most intervals slightly out of tune in comparison to their just ideals.
In this way, one could say that all classical musicians — pianists especially — play out of tune as a matter of course. (But, then, is it really out of tune...?)