They will occasionally develop some brown tips that can be simply trimmed off with a scissors. Most dracaenas are propagated by stem cuttings. On some varieties, the cut tip is rooted. On others, new shoots will develop from the old trunk. Ask for more information on a specific variety.
Brown edges are usually from a buildup of salts in the soil or a reaction to fluoride in the water.
Switch to distilled water. When it's over-watering, it happens gradually and the leaves turn a little yellow. When its under-watering, it can happen quickly and the plant recovers when watered. If it isn't just the natural shedding of lower leaves as the plant matures, it may be due to the plant being kept too warm and dry. Small or slow new growth usually indicates the plant needs to be fertilized more.