Hi there. The medal case isn’t supposed to be filled completely for a Progression reason. It simply displays the medals you have earned. You can’t hold anymore than the listed number. It is just a cap on medals, not a goal that you need to reach. It’s best to turn them in whenever our can.
I stopped playing this event as the medal rewards from cards etc. don't show up. I've got all the Director markers lit for depositing them but when I try to, the game says I don't have any and the cards are gone. No response on here from Bungie.