To be perfectly honest, we have no examples, either historical or archaeological, of ancient knowledge of fossils. There are carved mammoth tusks, but we have no reason to suspect that they thought these were mythical beasts rather than just large elephants. Some authors talk about giant bones in the context of marvels, but they mention them in the context of complete skeletons, sometimes with panoply. It is worth noting that they also talk about people who walk with their heads upside down, so we needn't take what they are saying seriously.
Giants were a pretty common explanation for those super-human bones, at least from the Middle Ages onwards. Although sometimes it was other mythical creatures like unicorns or griphons and sometimes even saints - especially Saint Christopher, who was already described as an impressively large man in Jacobus de Voragine's medieval "bestseller" Legenda aurea.
This is one of the most famous bones of the University of Vienna's geological collection, it's a mammoth femur inscribed "AEIOU", the motto of the then-ruling first Holy Roman Habsburg, Frederick III, and "1443", the year it was found. It was discovered during the preparations for the foundation for the northern tower of Saint Stephen's Cathedral and for a long time after were hung over the cathedral's Riesentor (literally "giant's gate", although actually not named after that bone) as a sort of public attraction.
Even much later the giant-theory still had a lot of traction. In 1723 brick factory workers at the Thurygrund, then north of Vienna now part of the 9^th district, excavated part of a set of giant teeth, which came to be known as Riese vom Thurygrund. Unfortunately these bones have been lost to disorganisation and mostly immediate destruction by brick workers but from drawings of two molars by its first descriptor, F.E. Brückmann, famous french zoologist Georges Cuvier later concluded they were most likely those of a woolly rhinoceros.
Wolfgang Lazius, Austrian humanist and official historian to Emperor Ferdinand I, writes about the supposed finding of the mortal remains of the biblical giants Gog and Magog. Unfortuantely, he is very vague so there isn't any proven connection to documented fossils.
After a storm in 1577 in the Swiss Canton of Lucerne monks of the nearby abbey discovered huge bones under a fallen tree, which they first assumed to be from a fallen angel, although natural scientist Felix Platter later concluded that it was merely the remains of an approximately 5,60m giant. And in 1605 people discovered a mammoth tusk in St. Michael (German Wiki link) in South-West Germany, which was interpreted as being the horn of a unicorn. To this day it's hung in the village's church and looks rather pretty.
A lot of short anecdotes, but I think this question pretty much called for it.
Sources:
This German article from the University of Vienna, some internet excerpts from Ernst Probst's "Das Mammut" and a few books by the great founder of paleobiology Olthenio Abel (mainly Die Tiere der Vorzeit in ihrem Lebensraum and a bit of Grundzüge der Paläobiologie der Wirbeltiere). Also, memory of many museum visits and Viennese folk tales.
This question was asked a few weeks ago, you can check out the few responses it got then, but more answers would be great!
I'm not sure if this touches on precisely what you're asking but it may help. The Chinese Shang Dynasty (1600 - 1045 BCE) practised pyromancy. They would heat up metal and touch them to ox scapulae or tortoise plastron to make cracks form. It was believed that reading these cracks would tell the future.
These bone fragments were cast aside afterwards and more than 150,000 of these fragments were found near Anyang in the 19th century. The villagers digging in the area initially used them as dragon bones and they ground them up into forms of alternative medicine.
