My personal experience has been that synthetics like nylon can pick up color more easily from other fabrics than natural fabrics like cotton or wool. But natural fabrics tend to be the worst for bleeding color to other fabrics and materials. Silk can sometimes be very, very bad, but it all depends upon how it was dyed and the process used. Some silk is fine, perfectly colorfast. (I'm sure Hermes scarves do not bleed. Haha!) Other silk no. For instance I recently purchased a silk blouse that had been sold under a designer label at Neiman Marcus. The dark green portions bled horribly onto the lighter portions of the blouse. But this is just my experience so I could be wrong about natural fabrics sometimes being more color unstable than synthetics.
I've been told by chemists who formulate fabric dyes that great strides have been made in the last decade in making fabrics more colorfast, in preventing color-bleed in the wash, but caution is still in order.