shot2 past and past
participle of SHOOT.
adjective
1 (of coloured cloth) woven
with a warp and weft of
different colours, giving a
contrasting effect when
looked at from different
angles: a dress of shot silk.
[New Oxford Dictionary of English]

Seems to have something to do with the definition for "shot" quoted above.
Looking at the definition, I don't think "shot" works together with "coloured". It would be a bit like saying "schillerfarben" in German.

Merriam Webster:
http://www.merriam-webster.com/dictionary/sho...
1
a of a fabric : having contrasting and changeable color effects : iridescent


From Google Books search on "shot colored", limit to 19th c.:

Annals of a Sportsman, 1885: He skipped in walking, moving his arms in a swaggering way, his cap over his ear, and he generally rolled up the sleeves of his military great-coat, which were lined with shot- colored calico...

My uncle Benjamin, 1890: Meanwhile my grandmother had put on her shot- colored silk dress, which she took from her drawer only on grand and solemn festival occasions; she had fastened over her round cap, in the form of a head-band, the finest of her ribbons...

The Universal Anthology, 1899: No more do I regret the mauve, delicate blue, purple, or shot-colored garments.

A search for this phrase yields many false positives, so I limited it to 19th-century books and found usage with both words together. So it seems to be something of a fixed phrase. The New York Times online archive has no usages from the 20-21 centuries, as far as I can see. I would have thought that at least their fashion reporting would have used the term. Is it still in modern usage? (I am unfamiliar with shot used in this sense.)



(This post was edited, after some more thought)

You still get shot silk these days, yes. You also talk about things being shot through with another colour. In both cases the 'shot' means one colour has been shot through another, with 'shot' as an adjective formed from a verb.
http://oxforddictionaries.com/definition/engl...

Shot-coloured is a bit different, as it would mean the colour of shot, with 'shot' as a noun, in which case it would presumably mean lead grey, as the noun 'shot' means 'tiny lead pellets used in quantity in a single charge or cartridge in a shotgun'. http://oxforddictionaries.com/definition/engl...