The most respectable long-term fallacy was actually the one that eventually brought the field to its current position, with Kenneth Jackson's radical theory (1955) that there were in fact two languages used concurrently by the Picts, one Celtic and one non-Indo-European. This is not a ridiculous suggestion, as even today there are many cultures which use two or more languages or dialects in the course of daily life (this situation is called 'diglossia' if you are more interested). But current thought has completely dismissed Jackson's idea, though it did help to bridge the old school (who are convinced that Pictish was non-Indo-European) and the new school (who believe that it's Celtic).
The solution the new school has come up with is that Pictish is a Celtic language that derived from an ancestor language called Pritenic, which shares an ancestor language with British (which itself is an ancestor of Welsh, Breton and Cornish) (Forsyth). So Pictish was a P-Celtic (Brittonic) language, but it was further removed from Welsh than Breton and Cornish are. Pritenic and British began splitting from each other probably in the 1st century AD (Forsyth). The evidence that scholars have used to support the argument for the 'proto-Pictishness' of Pritenic is based entirely on names of places, ethnic groups and individuals as found in Classical records (Forsyth). What this reveals is that most, but not all, are fairly clearly Celtic.
Much of the evidence for Pictish itself comes from similar sources, but there are a few other types of evidence, as well (Forsyth). Place-names and personal names are the most reliable sources, and are found in texts written by contemporary authors, such as Bede and Adomnán. Another type of evidence is found in the inscriptions made by the Picts, mostly on the famed Pictish stones. The Pictish king-list is also used as evidence, as it gives many personal names. Lastly, some evidence can be gleaned from the modern place-names of former Pictland. There are problems with all of these, but Forsyth asserts that they each support the idea that Pictish is Brittonic.
