The text of the alerts included "NYPD Twitter campaign 'backfires' after hashtag hijacked".
They were sent to millions of UK subscribers to BBC News alerts on iOS devices.
The BBC said it had not been hacked.
"We've been in the process of testing new functionality for our apps and a test message was sent in error this morning," a spokesman said.
"We apologise to our app users who were unnecessarily interrupted with the alert."
Push alerts for users of BBC News mobile apps, on iOS and Android, were introduced in July 2013.
App users receive alerts when there are major breaking news stories.
The BBC has apologised after sending two test breaking news alerts in error to BBC News app subscribers.