Sounds like maybe a file format issue. Unless you are talking about video you recorded with the phone yourself. Then it's not because of the file format. But if it's video you transferred to the phone, it might be. If you transferred the video file straight on to the phone internal memory or an sd card just by using drag and drop (or copy and paste) using Windows Explorer or synced it with Media Player, then it could be that the phone doesn't possess the correct codec to play the video file properly. That could result in stuttery playback, an out of sync soundtrack, or complete inability to play the file.
There are 2 ways around that problem that I know of. One is to use a 3rd party player off the Play Store like VLC Player or something like that instead. The other is to sync your video files to the phone with Sony's own Media Go software. If your video files are of an incompatible format for the phone it does a conversation process on any files you've selected to transfer, converting them into .mp4 or .m4a format or whatever it is the Z1 uses, before it actually transfers them and then they play pretty much perfectly on the phone after that.
If it's doing it with video you recorded or video you transferred with Media Go then that's another problem with the app or something and I would download another video player off the Play Store, there are a few there, and try that.
Did this help?