Yes, those were incredibly proliferate. Quilted cloth, gambeson, and in most cases where even metal armor is available, just layers of cloth would be used underneath it was commonly used whenever people could get their hands on it. But while it can offer resistance to a person with a pointy stick, sword, or axe, it is highly unlikely that it would offer very much protection against pointed projectiles without any other sort of stiff protection. A person would have to wear a lot of it for it to offer the protection a piece of treated plate the medievals used, to the point where it inhibits the person's movements.
There were some tests result made available on the Linothorax Project page. This link: http://www.uwgb.edu/aldreteg/AIA.Poster.LowRes2.jpg
That is very interesting, and I thank you for sharing it. I wished they would test bows above the hunting level range of draw weights though. 60lbs simply isn't even on the low end of bows meant for war. It may have been a contender during the ancient days when the Greek linothorax was around, though arguably they probably should have tested slings and lead balls rather than arrows seeing as they were far more common.
Further, someone should tell them to really stop using stationary braced targets when testing armor, it's not a good for any sort of armor that a human will be wearing, particularly if one of its main features is its ability flex with the human body when struck. For an incredibly high speed projectile, it may not be very significant as it will punch through before the armor or flesh can reach, but for arrows shot from a bow below 50lbs draw weight, this is going to make a difference.
Against Balkan, Turkish, Mongol, or generally any bow that goes above a 100lb draw force, cloth armor isn't going to do very much good. Take for example the emergence of the bodkin and similar arrow heads meant to deal with lightly armored or maille wearing targets. This is not to mention the much larger draw weights of crossbows in comparison to bows either. Quilted cloths, gambesons and the higher end linothorax style armors will, no doubt, provide a measure of protection, but I don't think it's enough to protect the wearer from the higher end projectiles like slings, war bows, and military crossbows.