There are three things that cause brown tips on Lucky bamboo:

1) Fluoride
2) Too Much Fertilizer
3)Dry Air

Fluoride should not be a problem since you are using distilled water. So it has to be a fertilizer or dry air problem. Change your water and don’t fertilize for a couple of months. Lucky bamboo like to be misted every day. Since yours is under a light which can remove moisture from the air either, increase your misting or move it from under the light. I would remove the leaves or trim the brown tips off. Your new growth should be healthy if you follow the suggestions above.

We have an page about Lucky Bamboo Care and a newsletter about Caring For Lucky Bamboo that might interest you.