While tartans are an expression of a specifically Scottish clothing type, and relatively recent at that, tartan-like plaid cloth has a much longer history. The Romans and the Greeks both made numerous mentions of the Celtae (and Keltoi) penchant for wearing "stripes" and "checkers" and other patterns that we would call plaid today—although the word didn't yet exist at the time. In salt mines near Salzburg, Austria, we've discovered (in 2004) very tartan-like textiles that date to the Hallstatt material culture from between the 8th and 6th centuries BC; a time where they are presumed to be at a relatively early Bronze Age stage of Celtic.
This, along with numerous other accounts, discoveries and whatnot, suggests that plaid is the ancestral textile pattern of all of the Celtic peoples.
But wait! There's more. A great deal of it, in fact. Similar finds come from Jutland and elsewhere in Scandinavia, which was never proposed to have been a Celtic area, and is seen as part of the core territory in which Germanic developed. Curiously, the area is more associated with the Anglo-Saxons who took Britain from the Celts than with the Celts themselves.
And even more bizarrely, the Chärchän Man, a naturally occurring mummy found in western Chinese Turkestan who died, it's believed, around 1,000 BC, is wearing what appear to be plaid leggings, not terribly unlike a number of pajama pants that I own.
Now, the Germanic peoples could have borrowed plaids from the Celts, and presumably borrowed other items of material culture and language from them, but the Tarim basin mummies are nowhere near the Celts, and frankly, predate them anyway. They're believed to be Tocharian speakers, descended from the Afansevo culture, which split off from the Proto-Indo-European the earliest—around 3,300 BC. Linguists, on discovering the Tocharian languages, were fascinated by some similarities to Celtic (as well as to Germanic, to Baltic, and Anatolian—but not with the nearby East Iranian languages) so much so that they initially proposed that some kind of Baltic or central European late PIE group had migrated across the entire face of Eurasia nearly to get to what is today well within China. Later, it was shown that Tocharian split off from PIE earlier than any other Indo-European language group except Anatolian, and the similarities are seen as archaisms that persisted among groups that had already spread to the "periphery" of the developing PIE sphere, while Baltic, Slavic, Greek, Armenian, and especially Indo-Iranian maintained a relationship that allowed them to continue to share isoglosses after the other groups had already gone their separate ways.
The intriguing part of this is the supposition that some cultural traits besides simply language can be traced all the way back to PIE unity. Some things, like the use of the word *h4eros or *h4eryos is the name for the entire group (found in all kinds of names, including Ariomanus, Eire (Ireland) Iran, and the term Aryan, etc.) can be pretty confidently assessed back that far, and a lot of work has been put into a three-way caste system, and various aspects of the shared, ancestral mythology. But can we say that plaid is the actual ancestral textile of our earliest Proto-Indo-European ancestors, and that it dates all the way back to nearly 3,500 BC in the late Eneolithic age?
I dunno, but I think I'm going to go buy another plaid patterned shirt, just in case.