There is screenprinting on Tie dyed shirts as a background.
Tie-dye - Wikipedia, the free encyclopedia
Otherwise hand-painting on shirts along with screenprinting makes no sense since screenprinting can print any type of artwork including using special paints for different textures and effects.
Hand-painted shirts would be sold with that as a selling point for people interested in it.
Just as shirts airbrushed by hand are sold as airbrushed shirts... not as silkscreened shirts with airbrushing on it.
You would be putting in twice the amount of work per shirt and it won't even look better than shirts using one method, it'll just be more expensive & time consuming to produce each one.
Maybe a famous artist can hand-paint his signature on silkscreened shirts as a limited edition but obviously not practical to print hundreds of shirts that way.
Traditional screen printing and hand-painted designs... 
