dont know what your plant is dont know access to gallery
but usually the reason plants give up the ghost apparently for no reason is root problems can you repot?
tom

Well, I've inspected the roots and there doesn't seem to be any sign of rotting or disease. I repotted in the summer and the plant really took off. I"m hesitant to repot again unless absolutely necessary. The plant itself is a tall, spindly thing, and the leaves started to fall off in the middle of the trunk. I can't figure out why.

Oohisay, I believe your plant is a succulent and not a tropical.
I have a similar plant, w/the commona name Joshua Tree, though not a true Joshua.
They require bright light, soil needs to dry between waterings, especially in winter. No fertilizer while plant is dormant. Misting isn't needed.
I'd treat as a succulent and see how it does..Toni

Hi,

Yes, I suppose this is a succulent, however I treat it exactly how you describe. I always let it dry, keep it bright, don't mist, etc. There is something wrong with the plant--nothing has changed in its care in three years.

do you have a county extension office near you? you could take the plant to them if it's manageable (size wise)? there are also master gardeners around if you search them out...maybe someone more local to you could examine the plant and see what they find....even an experienced nurseryman? worth a shot??