Downloading has never EVER worked in Jam with the Band on Flash Cards for some reason. I don't know if anyone's found out a reason for this, and as far as I know most other DS games download just fine when played through a Flash Card.
Getting custom songs is still possible, however! You will need BDX files and a song insertion program, though one of the programs you can get requires extra steps to make it work.
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Option One: BDX Tool
Note: Only songs inserted to the first 50 slots will be recognized. All others will be ignored.
Option Two: Degausser
Note: 150 songs may be inserted, but you must fiddle with the program and split your save file temporarily to make it work.
Option Three: Pre-Generated Save Files
Note: You will need to rename the save to be identical to the ROM, other than the ".nds" part
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By the way, if you preview any of the songs with either song-insertion program, they may sound wrong or out of tune. This is because a computer cannot play some of the sounds Jam with the Band can make, despite the game being wholly midi-based. Don't worry! When you play the song in the game, it should sound correct (there are a few rare exceptions to this, but the only way you can tell before playing it in-game is to find someone who's heard the song in-game, which can take much longer than just testing it in-game).