brake filament is a separate live but a shared earth so earth fault is ruled out at least.
bulb has 2 filaments, you should be able to see if one is broken. check the old and the new bulb (check the simple stuff first).
then clean the contacts in the bulb holder (next simplist to check).
if that doesn't work you'll need a multimeter as it's potentially a break in the wire, but I suspect the above will sort it.