The vast majority of leaf spots on palms are caused by fungi such as Cercospora, Cylindrocladium, etc. They are frequently more severe on juvenile individuals and those under stress.

The symptoms appear as small, round or oblong water-soaked spots varying in shades of gray, yellow, brown to black. They are often surrounded by a halo or ring of differing color tissue. The spots can increase in number or size such that many leaflets are infected and if severe enough, entire leaflets or leaves may turn brown, dry out and die. Plant stress, allowing the fungus to develop, may result from improper watering, sunburn, fertilizer burn, exposure to cold temperatures, or nutritional deficiencies.

If your palm appears to be root bound, up-grade it to a larger container, adding fresh potting soil. Try to keep the soil evenly moist and avoid prolonged periods of dryness. If indoors, house it in a location with as much bright light as possible, and if outdoors, avoid placing it where it is exposed to direct afternoon sun. Supply it with proper nutrition by carefully following fertilizer directions. To prevent further spread of the disease, you may also spray the leaves with products containing such fungicides as chlorothalonil or mancozeb.

Additional Resources
http://edis.ifas.ufl.edu/PP142