"Bulbils" are the little bulblets that grow in the leaf axils along the stem, as shown in Plantladylin's posting, 5 entries up from here. (Bulbils produce clones of the original plant.)
It you actually had these, then that would be important towards an ID. It points very strongly to tiger lily, Lilium lancifolium, or a hybrid thereof.
If you just meant "little offset bulbs forming underground", then it could be any lily. (However, since you refer to them "falling off", I assume your lily really does have bulbils.)
Easter lily, Lilium longiflorum, does not have bulbils, so if your plant has them, it would rule out this species.
