I solved a similar issue, where some songs in my itunes library would play to some random point and then skip to the next song. It started when I replaced my computer, so I copied over my library to my new computer and went through installing and importing on my new machine. When I noticed this behavior for random songs, here's what I found.
The affected files were .m4p files I'd purchased from the itunes store. This had nothing to do with activating them with my Apple account, which I did. The easiest way I found to identify the affected files was to view all your music in itunes, right click on any of the column headers, and select 'kind' so that column shows. Then order your music by the 'kind' column, and scroll down until you reach the songs where kind = 'Protected AAC audio file' (not to be confused with Purchased AAC audio file, which were unaffected in my case). Highlight just those 'Protected AAC audio file' songs, copy them, and paste them in some temporary directory. Back in itunes, now delete these hightlighted songs (I chose to keep the files rather than permanently delete them, but I think either would work).
Before adding these songs back to your library, if you want them pulled into your library directory rather than left in temp, make sure in Edit > Preferences > Advanced > check off Copy files to itunes media folder when adding to library.
Then in itunes, click File > Add file to library > browse to your temp directory > select all files > and click Open to add these files back to your library. After that, all my previously disfunctional files exhibited normal playback.
I hope you find this before you waste many hours on this like I did.
Good Luck, A.
