Can bacterial infection cause encephalitis?

Yes but likely viral: Encephalitis, inflammation of the brain, can be caused by viruses, bacteria, fungi, parasites, unknown causes, etc... Patients with identifiable causes usually have viral infections. Herpes types 1 & 2, epstein barr virus, varicella (chicken pox, shingles) virus, polio virus, coxsackie (hand-foot-mouth) virus, or mosquito-borne viruses can cause encephalitis.

Yes: Bacterial infections of the brain can cause encephalitis (there are several non-bacterial causes of encephalitis as well). A serious problem, may lead to death even with treatment. Patients with bacterial encephalitis will need to be hospitalized, intravenous antibiotics will be given, neurosurgeon possibly needed to drain infection.