To have access to additional Pastures, you need to put at least one Pokemon in every Pasture already unlocked. When that has been done, speaking with Marie (or whoever NPC can make you access the Pastures) again will give you new Pastures.
You can repeat this process several times to unlock even more.
The maximum number of Pastures that can be obtained with this method is 29. They can be unlocked at any point during the game, provided that you have a sufficient number of Pokemon to put one in every box.

Once the main game is completed, you can return to Marie to add 2 more Pastures to your holdings. This makes a total of 31 Pastures and 930 storage spaces.
The final number of Pastures seems to be 32. You unlock the last one by completing the Pokédex (catching them all, as confirmed by Joe).