If one's consumable inventory is full, then the Dark Fragments and the Offerings to the Oracle are not added to their inventory. Unfortunately, they also do not get redirected to the mailbox, so it seems that they are forever lost. This has caused me to become stuck at 2/3 Weekly Bounties completed for Petra, so I have to wait until the next weekly reset to complete my bounties, even though I have actually fulfilled their requirements (I just can't give the Offering to the Oracle Engine, because it disappeared).