From memory, I believe the Huns wore the silk not to stop arrows, but to aid in their extraction -- the silk threads wouldn't cut, so they could be pulled on to extract the arrows with a minimum of additional wounding. You could make padded cloth armor out of silk, that could conceivably stop arrows, but it wouldn't be normal clothing.
The Mongols did it. I can't remember if they used silkworm, or spider silks, but the silk would have possibly worked. You see, while it certainly goes in, the Mongol and Chinese arrows didn't slice the silk, so you could pull the clothing out around the wound and the arrow should come out. I canNOT account for certain arrows (Bodkins, for instance, or any culture that puts time making sure their arrows are sharp enough), nor for bolts (The additional force behind it might be able to flat out force it through). Either way, the damage is still done by the arrow, it just makes it possible to remove easily without killing or disabling the person who took the arrow.
here is a pic of an early chinese silk armor:
http://image57.webshots.com/457/5/28/60/2346528600100599642KJNzxn_ph.jpgit is quite padded, as far as silk doing that, most of the penetration would have to be stopped by armor first, since silk is quite similar to kevlar for the purposes of 'catching' arrows. it does not stretch much, thus if you have a shirt and it does catch the arrow on the way in, then at the very least it will either have to stop the arrow when it runs out of cloth, or break. it is going to need enough cloth in every direction of the wound for the penetration, hence if it penetrates 6inches (roughly short bow against unarmared) it will need to pull up the bottom of the shirt 6 inches (not too big of a deal), pull 6 inches of cloth from both sides (that means as much as pulling your shirt 6 inches out to the side which is pretty baggy for someone in combat, but not too unlikely if they wore armour over that), and finally pull 6 inches down (since gravity probably keeps the slack out this becomes quite unlikely). as for keeping out infection, i could see it catching the fragments from armour, but as far as contamanents go it would bleed through the silk, i could also see it as smoothing out the features of the arrow head allowing less damage from its shape, but these are all conditional on not penetrating the silk. the han tended to use poison tipped arrows when their shortbowmen were professional, thus the silk wouldn't do much good. the mongols used composite bows which would penetrate deeper than 6 inches without some substantial armour. the huns would have faced things other than shortbows in their war against the roman/gaul alliance far more readily than shortbows.
in the end i'd say that silk does have 'catching capability' but it would be quite doubtful if silk was ever worn because it can help one recover from a rather mild arrow injury, i would bet it's superior qualities as both insulation to the elements and its absorption of impacts were why it was worn, and most magical recovery properties attributed to it probably was because silk is far less likely to cause infection when penetrated than any other cloth used at those times.