Not necessarily. Keep in mind that the period clothing you've seen in museums are a couple hundred years old by now; of course they're probably faded from wear or exposure to light. Didn't that black (or red, or blue) chemically-dyed t-shirt you bought several years ago fade, too?

Some chemical dyes are more fade-resistant than other chemical dyes; and some vegetable dyes are more lightfast than other vegetable dyes. For instance, the blue from indigo (the active chemical, indigotin, is still used to dye denim) resists fading when exposed to light, whereas the blue from logwood is notorious for fading quickly. Moreover, redyeing old clothing to give it a new look was a common practice.