I've had this happen twice now; once in Haukke Manor and once in Wanderer's Palace. If someone skips a cutscene and then immediately leaves the duty (via the portal or the Duty menu) prior to the final chest being opened, it does not count as them "Passing" on the drop. What this means then is you have to wait around for the entire Loot Timer until it finally thinks everyone has passed.

First time was in Haukke Manor. A Lancer weapon dropped, but before opening the chest the healer left the instance. The other DPS left and I passed on the item (was tank). The Lancer rolled Need, but it didn't give it to him, so he had to wait until it timed out.

Second time in WP I Greed rolled on a staff. However, the tank exited prior to us opening the chest. Even though the other two DPS left (I was healing), it didn't award me the staff. I tried exiting, to see if that would force the issue and award it to me, but all it said was "Unable to obtain <item>."