This is a sad song about a former relationship. On Björk.com she says of this: "The first unhappy song I wrote was Possibly Maybe. That was very hard for me. Usually I write all the time, but that was like nothing happened for months. Then the song came out. I was ashamed writing a song that was not giving hope."
The song's video was directed by Stephane Sednaoui, who also provided the cover image for Post. Sednaoui recalled in an interview with The Guardian September 24, 2005: "I was supposed to be making this video, then I wasn't, then I was again. The indecision was due to the fact that Bjork and I had come out of a relationship with each other and the song was quite personal. Obviously, she wasn't sure if it was the right thing for me to direct the video but I convinced her that it would be good for both of us. The scenes of her walking in the sky were based on a dream she'd had. The shoot was very tense but not because of our relationship. It was because of what was going on in her life at the time: a mad guy in Miami had become obsessed with her and killed himself. The police found his body along with a written testimony that he had sent a bomb to her in the post.
During the shooting of the video she received news that the bomb had been intercepted in London. She was understandably freaked out because she realized it could have hurt her son. She was in a complete state of horror so we had to stop the shoot and give her a day to calm down. Perhaps in the end she used the emotion to good effect in her performance."