From what I see here:
https://projectpokemon.org/wiki/Pokemon_X/Y_3DS_Structure , each Pokémon is represented by 232 bytes, unless it is in the party, which adds 28 more bytes to it.
The total space that all Pokémon take is, thus: 232*30*31 + 232*6 + (232+28)*6 = 218712 bytes = 213 kilobytes (don't forget, 1kb is 1024 bytes, not 1000)
From what I see when I plug my SD card (which has digital versions of Pokémon games on it) , save files of Pokémon games are currently capped at 1 megabyte. Granted, this makes the Pokémon a little less than one quarter of the entire thing, but consider this: you need 1 byte for each eight event flags (by event flag, I mean, including every trainer fought and every item found), the Pokédex now doesn't simply use two bits per Pokémon anymore (one for seen, one for caught) ever since forms and gender differences were introduced, who knows how much is used for storing friend safaris (I'd say: at the very least 24 bytes per friend safari, since all custom strings are in multi-byte format, because they need to support different alphabets and stuff, so they have to use an Unicode-type encoding) , add to this all of the many features I forgot to even mention :) , include various checksums that are used to make sure the save file isn't hacked, forget about any type of compression (if the fact that the game takes a mere second to save is any indication) , and, at the end of it all, multiply everything by two: if previous games from Generation 3 onwards are any indication, there appear to actually be two save slots in Pokémon games, one is being used for the save proper, and the other is being used to store the previous save data as a backup in case anything goes wrong with the current one.
If the data that needs to be saved exceeds 250 kilobytes (the Pokémon data alone is 213) , that means that along with the backup, the full save will exceed one half of the allocated megabyte. Which means, that you can't have even two full save slots without cutting features and content that need to save stuff or sacrificing the backup feature.
After typing all of the above, I stumbled upon this:
https://projectpokemon.org/wiki/Pokémon_XY_Save_File_Structure . According to this page, one individual save file takes 420 kilobytes, and yes, there are two of those.
I might have overestimated the space Pokémon themselves take up, but my point remains more or less valid: there's way too much stuff for the games to save to include multiple save slots. Unless Nintendo makes carts that can save more that 1 megabyte of data, but I doubt they will do it just for this one game series or whether they can do it at all, I don't know what hardware constraints are in place.