I've been having this issue on all my PCs for years, but never on Android (where i generally listen to music) so I've never bothered writing about it before.
Basically, Spotify is unable to play the majority of tracks from one album in specific. The first track was either never added or removed from Spotify (presumably due to licensing / copyright business) and so I instead replaced it within all playlists with my own local copy of it. I do own the rest of the album too, however at the time I don't believe I set Spotify to use any of those.
Ever since, the desktop client has given the "Unable to play current song" message and skipped all but the one track I gave it, plus 2 which are also creditted to featured artists.
I have tried using local files in place of the Spotify-hosted ones for the remainder of the album, but it still skips and gives the error message. The local files themselves play fine in other programs. They are formatted exactly the same as the first album track, which does work fine in Spotify.
As stated above, all the problematic tracks play perfectly fine when I'm using the Android app, so I doubt this is an issue with the servers.