After surviving a stroke or event that results in traumatic brain injury, a very small number of people find themselves on the other side of the event speaking with a strong foreign accent. While it sounds made-up, foreign accent syndrome (FAS) is very real.
FAS is caused by damage to portions of the brain that affect speech and the coordination of the muscles used when speaking. The damage changes the way a person pronounces vowels and consonants, shifts speech rhythm and positions the tongue differently during speech. It wouldn't make you bilingual, but it would affect your speech to a degree that it sounds like that of a different country, or a region within your own country.
One misperception of this bizarre medical condition is that the affected person speaks in a different language or in a specific accent. It's not so much that you've adopted an accent, but rather that you've been forced to produce speech that sounds very similar to a pre-existing dialect. A person with FAS may seem to speak with traces of several different accents, and listeners may pick out the accent they're most familiar with. While no known cure exists, speech therapy produces some improvements for those with FAS.