Bangladesh playing first gave a target of 276 to England to chase. Bangladesh's Mahmudullah scored an impressive 103, followed by 89 by Mushfiqur.
But when it came to Englishmen, they were all back in the pavilion for just 260 runs in 48.3 overs.
England go out in the group stages for the third time in five World Cups, while Bangladesh advance past the first round for only the second occasion.
"I'm gutted at the moment," said England skipper Eoin Morgan. "There'll be an inquest over the next few weeks as to what happened and what went wrong. Then we'll go from there."
Read details at BBC Sport
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