Can a Nintendo DS Lite game fit inside a Nintendo DSi?

Yes, they can fit.

DS/DS Lite Slot-1 games are exactly the same as DSi Slot-1 games, but Gameboy Advance (GBA) games are not supported by the console.

Important:

Any Nintendo DS game which requires an addon that uses the GBA slot, is not supported on the Nintendo DSi (eg. DS Lite internet browser or Guitar Hero: On Tour)

Note:

Because of the above reason, do not sell your Nintendo DS Lite if you intend to play Guitar Hero On Tour. :)