Women Painters.com: Paint Issues and Fixes:

oor Scrub Resistance

Wearing away or removal of the paint film when scrubbed with a brush, sponge, or cloth.

image courtesy of
The Rohm and Haas Paint Quality Institute

Possible Causes:

Choosing the wrong sheen for the area.
Use of a lower quality paint.
Use of an overly aggressive scrub medium (see Burnishing).
Inadequate dry time allowed after application of the paint before washing it.

Solution

Areas that need frequent cleaning require a high quality paint formulated to provide such performance. High traffic areas may require a semigloss or gloss paint rather than a flat paint to provide good scrub resistance. Allow adequate dry time, as scrub resistance will not fully develop until the paint is thoroughly cured. Typically, this will be one week. Try washing the painted surface with the least abrasive material and mildest detergent first.