My guess is that there is something odd about those MP3 song files that iTunes does not like (and also affects playback on the iPhone).  No matter how it is added to iTunes library, or through your iCloud Music Library (for an unload), the problem persists.  When the song matches, iTunes does not have a problem with the AAC version from Apple's server because it's a completely different file.  But if you convert to AAC using iTunes, iTunes is using the source MP3 file with this issue.

If you open the MP3 file using QuickTime Player (right-click file and Open With -> QuickTime Player), does it play the song all the way through.  If it does, use the command in menu bar (of QuickTime Player) to File -> Export -> iTunes.  The song is converted to AAC and added to your iTunes library.  See if that version plays all the way through in iTunes.

Duplicate deleted.

so this problem did not occur with earlier versions iOS, so I must believe that the problem is specific to the iOS9 on my iphone, correct? Is Apple going to do anything about this. Over half my playlists are messed up by this problem, It is unacceptable. lew

My itunes library does this on and off and has done for a long time, playing the media file in another player works as you'd expect, the entire song plays.... it makes no difference it seems what the format is, mp3, apple lossless, AAC etc, in itunes library they just stop, and the same as you with my ipad and iphone.  I assumed its a library issue..... ultimately you can only really delete the file and start again but adding it new, at least thats what I did.  I've also discovered that creating a new iTunes library from scratch is an almost annual necessity these days.

I'd start by deleting and readding and see if that fixes it as a first round solution....

In your trouble-shooting, did you try deleting a song with this problem (after backing up its song file locally) from both your local iTunes library AND iCloud Music Library.  Then, in iTunes preferences General pane, turn OFF iCloud Music Library.  With iCloud Music Library turned OFF, add the song file back to your iTunes library.  See if it plays properly.  If it does, turn ON iCloud Music Library, and see if it continues to play properly.  The problem, if this works, is that you will lose the supporting metadata for the song, like date added, play count, rating...  And you'll need to add the song back to any playlists that previously had it.  But your iTunes library (and iCloud Music Library) should treat it like a new song; if there is nothing wrong with the song file, it should play properly like any other song file you add.

Suggestion to export from Quicktime player was extremely helpful, and first solution that worked for me. Thanks!

Glad to see some people read the old posts...   Thanks for replying back.