For a band I hate, Keane.
For a band I love, how about Antlers.
Their album Hospice is a concept album, and "tells the story of a relationship between a hospice worker and a female patient suffering from terminal bone cancer, their ensuing romance, and their slow downward spiral as a result of the woman's traumas, fears, and disease. The story of her deterioration also serves as a metaphor for an abusive relationship."
The music and lyrics entirely live up to this description. As a bonus, they also songs on other albums called "Every night my teeth are falling out", and "Putting the Dog to sleep".
Highly recommended :)
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As a bonus, I'd add in Aldous Harding, who has a song called "What if birds aren't singing... they're screaming" - that's surely one of the most depressing thoughts in pop.