Personally, I always attributed Harry's realization of which relic Nico truely wanted to the shroud, not the knife. Nico had just killed his own daughter, and it's hinted that he knew what was in the vault before the heist. While I honestly doubt it would have worked after what he pulled... I can't help but wonder if Nico's plan was to get the items, get away with his daughter's corpse, bring her back, and be in possession of all the relics*. I don't have any substantial support for this theory other than Harry stuffing the shroud into his duster pocket**, but... *shrug*
It's just a gut feeling.
* I support the "use in a summoning" theory, for these. Though, who I believe they are meant to summon, I'm still undecided.
** Correct me if I'm wrong here. I seam to remember this being the case. But, I don't have hard or softcover versions of the books. I only listen to the audiobooks. As a result, checking for exact details is difficult.