Your cane plant has the botanical name dracaena.These plants are native to Africa and Asia. A tall dracaena may resemple a palm tree but have quite different cultural requirements from classic palm trees. In particular, dracaena are able to manage with limited bright, indirect light of just a few hours a day, so it is probably in the right spot.

In addition to the browning, new growth that you mentioned, I notice in the photo that the leaves are drooping a bit more than they should. That could be because it is experiencing cold drafts. Is there air conditioning in your home? Dracaena like to be kept at 65 to 75 degrees. If the temperature drops lower for a week or so, the plant will recover but only if temperatures above 65 degrees are restored. If a plant is kept near the vent where cold air is coming from a cooling system, it is experiencing intermittent blasts that are far cooler than the room.  Your plant also needs a great deal of humidity. Frequently spraying the leaves if they are reachable, operating a humidifier or keeping dishes of wet pebbles near the base of the plant will add to its good health and reduce the chance of insect problems.

The brown new growth you are concerned with could also be related to blasts of colder air. Stem rot begins at the top of the plant and moves downwards. It can only be addressed by cutting the plant back dramatically and waiting for new growth to sprout.

If you are sure that your plant has not been exposed to cold air, the problem could be over-watering. You did not mention whether leaf tips are also turning brown, but that would probably also occur if the plant is getting too much water and some of the leaves may be turning yellow and dropping. Your plant needs to be kept moist during the growing season but never standing in water and the soil needs to drain well. If you think this could be the problem, you will need to un-pot the plant and check whether the roots have rotted to determine if the plant will survive. If a good portion of the roots are still white and healthy, you can cut away the diseased roots and re-pot.

Courtesy of NYBG Plant Information