Cloak of the Arachnida does not appear to make one immune to web, as it is listed in the item description.
That report is too vague to work with. Can you be more specific about which version of web (the regular spell, the Shadow Conjuration version, a spider's web ability, etc.)? If it's a spell, who cast it? At which point did it not work?
Background:
In my experience it doesn't works with the shadow conjuration version of web at all (probably mechanically correct), and it works with the conjuration version about 50% of the time. When I used to use the cape more often, I would have it equipped already when I logged in. I would then cast a conjuration web, and then walk into it (usually while fighting). Half the time the cape worked, half the time it didn't work. Unequipping and re-equipping the cape at this point did not reset the effect and make it start working.
When it works it always works for that session. When it doesn't work it always doesn't work for that session. So my guess is that if you login and dont see the bug, you are probably not going to see it that session.
I'd include a screenshot, but after my last level I removed the spell from my spellbook, so I can't create the scenario anymore. Maybe someone else can include a picture.
Reproduction steps:
My guess is if you have a character with the cloak that can cast web. Log in every so often and cast web, and walk into it. You should see the bug after a few tries.