The Rembrandts’ ‘I’ll Be There For You’ may be one of the world’s most instantly familiar TV theme tunes, but its success split up the band, amid fears they’d lost everything about them that was cool.

No one told them the song was gonna blow up this way. The Rembrandts were putting finishing touches on their third album, “L.P.,” when a sidestep into television drastically altered their course in ...