A swimming pool should have about 2 parts per million (ppm) of chlorine to ensure sanitary water. Swimmers are also able to swim with comfort in pools with chlorine levels as high as 4 to 5 ppm, though it's not recommended. Such low pool-chlorine levels can't come close to causing colors in clothing to run or fade. Bleach itself contains over 200 pm of chlorine per tablespoon, and it would take roughly that to begin to immediately bleach clothing.
The formula for bleach when it's used to disinfect pool water is 1 gallon of bleach per 30,000 gallons of water. This will raise a pool's chlorine level to 2 ppm. It also takes about 1 lb. of typical granulated pool chlorine per 30,000 gallons of water to achieve a 2 ppm pool chlorine level. A pool's large amount of water effectively dilutes chlorine, thus preventing it from rapidly bleaching clothing.
People love to use outdoor swimming pools when it's very sunny, which means clothing that ends up in a pool is also exposed to a great deal of sunlight. Naturally, there's a tendency for pool water-soaked clothing to fade over time and with exposure to the sun. However, such fading is due to the effects of the sun, not the chlorine levels in the pool.
Elevated pool-chlorine levels really can cause fading of clothing over time. Also, vinyl swimming-pool liners and even colored pool tiles exposed to constant high levels of chlorine will bleach out over time. Keeping swimming pool chlorine at about 2 ppm, though, can prevent most pool bleaching from occurring.