PokéBase - Pokémon Q&A
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The player can withdraw or deposit Pokémon in a number of Boxes, allowing them to own more Pokémon than just the six permitted in their party. When a Pokémon is caught but the player's party is full, the Pokémon will be sent directly to the last Box that was accessed on the PC; from Generation III onward, if this Box is full, the Pokémon will be placed in the next free Box;

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The Pokémon Storage System is typically accessed through PCs, which are typically found in Pokémon Centers, Day Cares and Nurseries, and battle facilities.

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Why do I seem to get the most votes for answers that were easiest to copy-paste from Bulbapedia?