We have had this plant for some time and have used fungaside and pest killer on it but these do not seem to have worked. Dracaena with brown/black leaf edges ( photo / image / picture from bigbadmur's Garden ) Dracaena with brown/black leaf edges ( photo / image / picture from bigbadmur's Garden )
The most likely problem is overwatering. Check to see if it is sitting in water. It should have a pot that drains well. Next I would look for a source of dry heat like a forced hot air heating register. Jerry
I don't think your plant has any pests. Looks like it needs to be repotted, combined with fluoride toxicity. Dracaenas are also sensitive to lime, chlorine, tap water can really make them look ratty. Rain, distilled, and condensate from a dehumidifier or A/C are sources of water w/o these substances. If you find a pancake of roots at the bottom of the pot, saw that off. The old soil should come off much more easily then. Remove as much as possible, then replace with new soil, so that it comes to the same level on the trunks as it did before. Will likely fit back in the same pot very nicely. One can help prevent fungus gnats from taking up residence in the soil by keeping the surface clean of dropped leaves.