1. In the Web Player, go to "Your Music" -> "Songs"
2. Play songs as normal
3. From the "..." menu for a song, pick "Remove from Your Music" (when I did this, I had previously selected to play this song, not sure if necessary to reproduce).
4. Any songs above the deleted one in the list play fine.
5. Any songs below the deleted one in the list start playing the song above them by 1.
6. Something that could be tested: delete more songs, and play songs further down in the list. I suspect that Spotify just doesn't update the internal array of songs properly (though it does update the visual one).