>>UPDATE: apparently "tatty" is a dictionary word for "shabby/dilapidated".
I'd say that "tatty around the edges" often refers to something that used to be a favorite but has become worn through use. Often associated with longing and/or with nostalgia- I'd like to use it but it's a bit tatty round the edges.
"... They ne'er cared for us
yet: suffer us to famish, and their store-houses
crammed with grain; make edicts for usury, to
support usurers; repeal daily any wholesome act
established against the rich, and provide more
piercing statutes daily, to chain up and restrain
the poor. If the wars eat us not up, they will; and
there's all the love they bear us."
-- Shakespeare: Coriolanus, Act 1, scene 1
