Many people have posted about issues playing music that was supposedly downloaded since iOS 9.2. I have had some issues - but generally the system works as expected. I even managed to get a 500 song playlist downloaded properly which I was never able to do before 9.2.
(1) First suggestion would be to try either a force quit of the Music App and see if that helps.
(2) Would be do a hard reset of the iPhone (holding HOME and POWER buttons until the Apple logo appears)
(3) Next suggestion would be to remove whatever songs/albums you think are on the iPhone. Then go to Settings/General/Storage & iCloud Usage/Manage Storage and delete any data that is being used by the Music app. Then - after restarting the iPhone - try downloading the music again and see if it works properly.
(4) Just wait (somewhat patiently) till iOS 9.3 because supposedly one of the fixes included is about the downloaded songs in the Music app.