I had my water tested at a local pool store to test the accuracy of my home test kit. Here are the results:

Free Cl = 3 ppm
Total Cl = 3 ppm
pH = 7.6
Alkalinity = 90 ppm
Calcium = 220 ppm
Cyanuric Acid = 80 ppm
Phoshpates = 1000 ppm

I was expecting to see a much higher CYA reading, but the tech that helped me said that this level (80) isn't cause for too much of an alarm. The phosphate reading was very high, so I added a "Phos Free" product that they sold.

The high phosphates may explain why I occasionally get a light algae on the side of the pool even though I have Cl in my system, but I don't think that it explains why my Cl readings are always high.