If two binary brown dwarfs began to exchange gas with each other, would they both ignite into stars? If two brown dwarfs got close enough to exchange gas with each other, could it trigger nuclear fusion within their cores? Is it possible for two brown dwarfs to orbit so closely and become a rocheworld in the first place?
It is possible for two brown dwarfs to form a contact binary. This would not, of itself, cause core temperatures in either to rise. 

If the two brown dwarfs were to merge, the mass of the resulting body could be enough to generate sufficient heat and pressure in the core for nuclear reactions to start. If both brown dwarfs had a mass of about 60 Jupiters, the combined size of 120 Jupiters would certainly be big enough to form a red dwarf.