Only 30 cells were offered by the system to gain bags when I entered into neverwinter. Later, I also found that my bag place is too narrow along with the numbers of my loots getting more and more. I was wondering at that time about how I can expand my bag size. After playing the game, I concluded several ways to gain bags in the game.
the first way to gain neverwinter bag
Mission: Close to the Crown
This is the 8th level mission. Adventure's Knapsack will be gained after completing the mission. And the place to do the mission is Blacklake District Quests. An accessible door would turn up around you when you finish your mission. You'll guide a Archer to look for the Karzov's Guards and then gain your first extra bag which occupies 18 cells only on the condition you defeat them. No worries about the size, there's also another way to gain bag.
the second way to gain neverwinter bag
Mission: Clockwork Guild Tomb
The place for achieving bag is in the Nevedeath Graveyard Quests. The adventurer's pack will be achieved after completing the mission. This is an interesting mission. You'll activate several huge clocks in the game. Only after that you can go to the bottom of the dungeon to kill Shoth the Tomb Queen, which represents the end of your mission.
Other non-task ways to gain bag
1. to purchase Founder's pack which send you package as the gift
2. to purchase ZEN which can help you buy bag in the game shopping mall
3. to purchase bag by spending neverwinter astral diamonds through auction house
All the ways mentioned above are what I know to purchase bag currently. If you have a lot of astral diamonds, the fastest way to gain bag is through the auction house.