I'd read the destructions very carefully for fluid compatibility. The problem may be the left over DOT 3 in the system, it is doubtful you can get it all out. And are your seals and other rubber (generic term) parts compatible with the new fluid?
I remember when DOT 4 first came out, it was supposed to be the answer for many of the issues with DOT 3. It may have been the answer, but it wasn't very long before many people discovered DOT 3 and 4 aren't compatible and shortly thereafter they came out with DOT 5. And shortly after that, many car manufacturers went back to DOT 3.
You can't always believe the claims by the manufacturer of the fluids, as far as fluid, metal, and rubber (generic term for the synthetic blends used in seals, "O" rings or whatever) compatibility is concerned. Trouble may not show up for years.
I've purged brakes line and what came out looked kind of like metallic cottage cheese. Fairly obvious somebody put the wrong fluid in there.
DOT 3 isn't perfect but it works well enough. Just make sure you use a closed container that has no possibility of having any moisture in it. And purge the old fluid out periodically with a name brand replacement. You never really know if the budget stuff is just the premium stuff relabeled (generic) or a blend by a less than reputable company.