Dracaena plants are fairly uniform in their cultural requirements - bright or filtered sunlight, consistent moisture at the roots, a monthly feeding through the growing season (March through October) and adequate humidity in the environment.  

Lack of humidity is one of the most common reasons for problems with dracaena.  Leaf tips turning brown accompanied by leaf loss and yellowing leaf margins are indicative of low humidity.  Placing your plant on top of a dish filled with small stones and water will improve plant humidity. (Do not allow the water to touch the bottom of the plant pot).  Spray daily with a mister; consider the use of a humidifier in the driest months.  

Exposure to a source of cold air (open window, air conditioner) can produce the same effect.  If your plant is right at the window, the cold nights may be adversely affecting it.

It is less likely given your plants appearance that watering frequency is an issue, but you will be the best judge by examining the brown areas carefully. Brown spots and crisp edges of the leaves accompanied by leaf curl and stem weakness are symptoms of underwatering.  This seems less likely to be the problem with your plant as the leaves appear generally upright and sturdy.  If the spots on the leaves are soft and there is any rotting, wilting or bleaching, you are overwatering.  

There is also a bacterial disease called leaf spot that may be present if overwatering, chilling and insufficient light are affecting your plant.  Look for damp or blistered areas on the leaves to determine whether you have leaf spot.  If so, remove affected plant material and increase temperature, light, ventilation and reduce watering frequency.

You should also be aware that dracaena leaves yellow and fall off as they get old so you will see the lower leaves deteriorate as they get to be a year or so old.  

You can find more information on the care of this plant in our Guide to Dracaena.

Courtesy of NYBG Plant Information