It ll depends on the specific dye used, and how much is used. There are many red dyes that do not bleed at all, but they cost more when purchased in industrial quantities. When clothing manufacturers cut corners, they use the cheaper dyes which bleed. If they use a small amount of the cheap dye to produce a pastel color, there's less dye available to bleed.
Some clothing manufacturers are very reliable about using only washfast, non-bleeding dyes. None of the red clothing I've purchased from Land's End, for example, bleeds in the laundry.
There is a commercial dye fixative, called Retayne, which you can use at home to set the dye in purchased clothing. It works on any kind of dye except for indigo (the blue dye used for jeans). If you have a quilting supply store nearby, or an exceptionally good sewing store, they might carry Retayne, or you can mail-order it from any good dye supplier from one of the companies listed in the links below.
If you are dyeing your own clothing, you will find that Rit and Tintex all-purpose dyes bleed badly in the laundry, unless you use Retayne or a similar brand of commercial dye fixative. All-purpose dye will continue to bleed in the laundry if you try to fix it with salt of vinegar; they don't work at all.
In contrast, if you buy a good cool water dye, such as Procion MX dye, Tulip One Step Fashion Dye, or Dylon Permanent dye, you will find that no dye bleeds at all, once you've removed the excess unattached dye by washing twice in HOT water (140°F or hotter). Quality really counts, when it comes to dye choice. Oddly, the good dyes do not cost significantly more than Rit dye, in local shops. It's only in larger quantities that the poor quality dye costs less.