Make sure neither Shuffle or Repeat are switched on in the playlist, either of those can prevent you reordering tracks. Also make sure you click on the number column (extreme left). Click on the song you want to move, hold down the mouse button and drag the song to where you want it to be.
No dice. Shuffle and Repeat are both turned off. I click on a song number — never releasing my finger and try and drag a song and when I release my finger from the mouse the song just remains where it original was.
Well, I finally got the results I wanted by simply deleting the playlist that didn't allow me to move things around and created a new playlist. Now everything is hunky dory. Thanks for the tips, everyone.