There is an extensive discussion of the protection provided by armour in section 9 of Alan Williams The Knight and the Blast Furnace (Brill, 2003). This focusses on iron/steel armour, but reports some results for non-metallic armours.

Longbows and crossbows can deliver 200J of energy at short range. To stop this with plate armour, 2mm (backed by a few layers of cloth) should suffice, if the iron/steel is of good quality. Unsurprisingly, late Medieval breastplates of 2-3mm in thickness are common (older thinner Medieval breastplates may have been intended for wear over mail).

From the experimental tests reported in Williams, 30 layers of cloth will stop an arrow with about 100J of energy (comparable to 1-1.2mm of iron or unhardened steel). To stop arrows from high draw weight longbows and crossbows, one would want 60 layers.

Noting that 30 layers of linen will weigh about the same (per unit area) as 1.2-1.5mm of iron/steel, arrow/crossbow-proof textile armour will weigh somewhat more than equally-protective iron/steel plate. A breastplate of about 2mm thickness will weigh about 2.5kg, so similar protection for textile armour for the front of the torso, neck to waist, will weigh about 3kg (and be much thicker). A complete gambeson with such protection for the front of the torso will be heavier.