Ha, ha. Well, a townie (old timer, family been around for years) might pronounce it "Bedfid," but I can assure you that SheFly (who lives there) and my friend Jack, who also lives there, do not! And, my son lives in Medford and says, well, Medford. Some of the older people there actually say "Meffa,"
When i was a kid, I talked that way, but 23 years away, cured me quite quickly. Around where I live, very few people have a Boston accent. When I first moved back here, I lived a bit north of here, near Lowell, and the accents were very strong. I once had one of my kid's teacher write a note talking about the drawer, spelled "draw." That's how she said it, and I guess getting a master's degree didn't teach her any differently.
The only time my accent comes out is when I get angry; when I yelled at my kids they would make fun of me. It's still not natural for me to say "r's" but I don't ever slip up.