It is possible to dye bleached hair a brown color, although it is important to wait a few weeks after bleaching the hair before dying it to minimize damage from over-processing. Those looking to dye hair will need to take existing color into account when choosing the shade desired because that will be the starting point when it's time to color.
As a general rule, hair can be dyed up to two shades darker or one shade lighter if the person has previously dyed the hair before. Those with hair that has never been dyed can easily go two shades lighter or darker with no problem. It can be a good idea to remove any existing dye from the hair with a stripping product prior to applying color for better results, especially if your hair is naturally darker than your existing dye job. Stripping agents are available at most big-box beauty supply stores for those attempting an at-home dye job, and all hairstylists should be able to remove the existing dye from the hair before applying the fresh dye. In fact, most will recommend it. A toner following the coloring process can help tone down darker pigments when making a change from really light to darker colors.