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You could also try Settings then tap the 3 dots then tap reset app preferences reboot and test
Open the messaging app > menu > settings > check under notification settings.
Have you installed a third party messaging app and/or notification app?
Follow @Phy 's advice, boot on safe mode, and check again, if you receive notification while on safemode then its a third party app acting up
You could repair your phone with PCC
Bridge (for Mac)
Alternatives on How to backup Xperias
http://talk.sonymobile.com/thread/36355