There is actually more than one adapter with that model name. To find out which one yours is type into terminal:
lspci -vvnn | grep 14e4
My netbook running 14.04 has a broadcom 4313 with 14e4:4727 (rev 01), and what I've done with this adapter (which isn't nearly as pesky in 14.04 as before) is for 4727.
Definitely purge bcmwl-kernel-source as mentioned if you went into "additional drivers" ... unfortunately this isn't always reliable. I don't use additional drivers anymore.
I installed linux-firmware-nonfree, which was a mistake on my part. If you installed it yank it.
Follow the steps here:
I've had lubuntu 14.04, xubuntu 14.04, and now LXDE (14.04 based) and it's worked every time.