This happen to me with a video. The audio was fine until about 10 to 15 seconds at the end of the clip. The video/audio played completely in VLC and Quicktime. To fix this I convert the audio from the video to mp3, add it to Premiere, link it with the video and all was good.
Badmonkey: You see the waveform from the audio suddenly going completely blank, right?. I believe that the mp3 format could be damage in some songs. Try converting the audio to wav, aac or aif and see what happen. You can also convert the audio to video (this is not that practical, but it works) and then in Premiere simply remove the video.
Robertcfi: What tool are you using to move your audio/video on the timeline? If you use the selection tool to move your audio/video it will move that specific clip that you selected. If there another clip in front or behind the timeline and you move your video over another video it will be overlay (It will delete the video/audio bellow) You can hold the control key on PC (Command on Mac), drag the video clip anywhere and it will overlap any video/audio (the video/audio will move to the right or left) I use the track selection tool to move clips in the timeline. It moves the clips from one video and audio track (be careful because you can overlay clips) If you hold shift you will see two black arrows, this will move all the video and audio tracks.