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Try this: settings-apps-all,scroll down to movies (video),force stop and clear data.
Usually 1080p videos might lag a bit when you fast forward, the processing takes some time.
transfer them to a computer and try to play them there, just to double check that the files aren't corrupted.
Boot on safe mode and try again
http://talk.sonymobile.com/t5/FAQ/How-to-boot-your-phone-into-safe-mode/m-p/348008#U348008
If what uli suggests doesn't help,you may need to perform a software repair using PC Companion.The process will erase all internal memory.You might need to charge your device for about 1-2 hours before so the device has at least 80% battery.
Make sure the device is not connected to computer or laptop and turned off (sometimes you have to press power button and volume up until the device vibrates 3 times or on some devices you have to press the small button near the SIM tray or on the back of the device).
Open PC companion->support zone->phone/tablet software update->repair my phone/tablet blue highlighted.Follow the steps given,do not connect your phone until prompted to press volume down while plugging USB cable. If PC companion does not start the repair process, you might need to run it in compatibility mode with Windows 7 (in some cases XP).
Anytime,this was a strange case indeed.