You probably do not have a keyframe at the 3 second mark. Because non-keyframes encode differences from other frames, they require all of the data starting with the previous keyframe.
With the mp4 container it is possible to cut at a non-keyframe without re-encoding using an edit list. In other words, if the closest keyframe before 3s is at 0s then it will copy the video starting at 0s and use an edit list to tell the player to start playing 3 seconds in.
If you are using the latest ffmpeg from git master it will do this using an edit list when invoked using the command that you provided. If this is not working for you then you are probably either using an older version of ffmpeg, or your player does not support edit lists. Some players will ignore the edit list and always play all of the media in the file from beginning to end.
If you want to cut precisely starting at a non-keyframe and want it to play starting at the desired point on a player that does not support edit lists, or want to ensure that the cut portion is not actually in the output file (for example if it contains confidential information), then you can do that by re-encoding so that there will be a keyframe precisely at the desired start time. Re-encoding is the default if you do not specify copy. For example:
ffmpeg -i movie.mp4 -ss 00:00:03 -t 00:00:08 -async 1 cut.mp4
When re-encoding you may also wish to include additional quality-related options or a particular AAC encoder. For details, see ffmpeg’s x264 Encoding Guide for video and AAC Encoding Guide for audio.
Also, the -t option specifies a duration, not an end time. The above command will encode 8s of video starting at 3s. To start at 3s and end at 8s use -t 5. If you are using a current version of ffmpeg you can also replace -t with -to in the above command to end at the specified time.