no, you can't do that. DSIware programs are short little games for you to play that you can download. I don't think you can put DS ROMs on your DSI and they would magically work. But I DID find A Thing called the iPlayer which allows you to play GBA on your DSi. www.youtube.com/watch?v=cO-mMMycf_0
If you want to play DS games, i suggest you get a DS game cart.
Half of what Aijam said was true but im not sure if this will work but some ppl have flash cards for the ds what enables you to play any game whats a rom in J K EU U ect its basicly a emulator but hand carrying on your ds.
Flash cards lets you download a game and put it into a small micro chip what goes into your ds game (well thts mine)
The ones i know that work are SuperAce i think EDGE R4 and thts all i remeber and i have EDGE xP