Looks like a Draceana (closely related to Cordyline). Do you let it dry off between watering? The lower leaves being the oldest will normally die back first. This is natural over time though it can overtake new growth if conditions are not correct.

When was it last repotted? The compost may be old and manky by now. Buy a fresh bag of compost for houseplants and give it a treat!

It looks to me like one of the newer Cordyline hybrids involving C. terminalis, but the treatment is the same:

Give it lots of indirect light--within 3 feet of a north window with no overhang is ideal--but no direct sunlight. Allow the top 1/2 inch of the potting soil to dry between waterings, and keep the saucer underneath dry--either take it to the sink to water it, then let it drain, or empty the saucer with a bulb baster after watering. Use any good houseplant food, but only 1/2 the recommended amount. If you can raise the humidity, that will reduce the browning of the leaf tips. Inspect regularly for pests, such as mealybugs or spider mites, and treat accordingly. If you repot it--which it doesn't seem to need now--look for cactus mix, since it loves fast drainage.

Hope that all helps, Mirya!

I just found this link again after all this time. Thank you for the information! Cactus soil really does work well!