My hair is naturally black. About a month ago I bleached it to a yellow color and then dyed it over with Splat "Blue Envy" dye. It has had some time to grow, and I've done over the roots with bleach and dye again several times, and each time (including the first) about a centimeter of roots have remained a coppery-brown. This is immediately after washing out the dye, not from fading over time.

What am I doing wrong? The rest of the hair looks fine, not even fading. I thought maybe the hair at the roots wasn't bleached enough to hold color, but I only bleached my tips to dark brown (much darker than the roots) and they are holding the color fine.

The ends are probably more porous and therefore hold the dye more easily. The only way you'll get the dye to hold well at the roots is to bleach them a bit lighter.

I had the same problem with some chemical purple dye that lightened and dyed hair at the same time. Wrapped the highlights in foil and always the top 1 cm turned out blonde while the rest was purple. Had me puzzled, since I applied the same amount of dye and wrapped it all the same. Since this Splat is not mixed with peroxide though, I doubt it's the same case.

Did you apply on freshly washed hair? 'Cause otherwise the skin's oil may prevent the dye from sinking in the nearby hair. Else you may indeed need to bleach lighter.