Can you put movies on Nintendo DSi?

Yes, it can be done very easily at less expense than buying a psp. There are a few options available one is the iplayer for dsi/ds found atwww.dsiplayer.com and the gba player compatible with dsi/ds and gameboy advance www.ps2cover.com/gba-movie-player.shtml

all software is included with these two options, all that is needed is a micro sd storage card and card reader to transfer files from your computer

The third option is the acekard 2i with requires homebrew software such as moonshell that will need to be downloaded to your storage card as well as conversion of the movies to a usable format, a micro sd card and card reader. The quality of the homebrew software is of a slightly lesser quality than the commercially available setups.