Think about the size of the game world, all those little detailed sprites, every single map, and everything that would possible have to be saved on that tiny card in order to run the game.
Now think about the amount of space that you would need to store everything about the player,
For each Pokemon:
-it's current health
-stats
-level
-what attacks it knows
-how much pp it has for those attacks, remember a pp up can cause a non-default number of times, meaning the game has to actually store it.
-its personality
-how much exp to the next level
-date, time, level and location of capture
and that's just some of it; each Pokemon has an incredible amount out small stuff stored with it. That adds up to a colossal amount of data when you have even over 100 Pokemon.
Now add to that your player's
-location
-name
-Trainer ID
-badges earned
-what trainer's on the map you've beaten
-how many steps you've taken since you beat that trainer (for rematches)
-what Pokemon they have in their party
-every single freaking item you are carrying, remember originally your bag and PC had storage limitations, they don't anymore so you can stuff hundreds of items in your backup, (hammerspace ftw)
-story progression
-side mission progression
-which pokemon is in what box in which particular of the 30 spaces
-box name and theme
As well as any multiplayer and online data that you may have and the National Pokedex.
Obviously I didn't list everything, but think for a second about how much that data that is, and think about how big much room that game card has on it, anywhere between 64mb and 4gb, that's not a lot of room even at 4gbs for all the music, sprites and engine data as well as the giant save data file. Even the amount of time it takes to save can be attributed to how little memory is in available in the standard DS and even the DSi and 3DS, less than a computer in the early 90s.
While I do think that Nintendo will never increase the number of save files past one, even if the storage becomes available, I do think it is still a matter of not having enough space leftover for another potentially massive save file, you would probably get a lot of 'out of space' errors if you had more than one. Keep in mind a lot of games that have more than one save are usually:
a) Not very large games or
b) Don't use 3D models or don't have the level of detail that the Pokemon games do, or
c) Don't save all that much data
Even if I'm wrong and it there actually is enough space on the card, anyone here who simply dismissed it as a way for Nintendo to sell more copies, I don't think you respect how much these games truly keep track of in order for people to get the experience they do. I wouldn't be surprised if it stores data that we never even see, but it needs for engine and gameplay purposes.
EDIT: The guy who said the same thing above me posted while I was writing mine, sorry if they say the same thing ^^;