From the “Findings” column (by Rafil Kroll-Zaidi) in the January 2011 Harper’s Magazine, this note (no source provided; relevant expression boldfaced):
Discus fish rear their fry by allowing them to feed continually on their own skin mucus for the first few weeks of life, but whenever the fry have bitten one parent for about ten minutes they are sent off to bite the other.
The writer presumably thought that their by itself allowed for two choices of referent, one picked out by discus fish, the other by their fry and them, and that adding own would disambiguate things in favor of the first. But in fact it does no such thing, and to my mind actually tends to favor the second reading (though in either case the following clause clarifies whose skin mucus is being nibbled on).
Their parents’ instead of their (own) would be a considerable improvement, though it would still allow for the very unlikely reading that the fry are allowed to nibble on their grandparents’ skin mucus. With pronouns, there’s rarely a way to exclude unlikely readings.
Now, own can be a useful tool. In describing the sexual encounters in gay porn flicks, I’m often faced with complex choices of referential expressions. Here, for instance, is a portion of the summary of some action in the film Manifest (full posting here), involving the actors S and W. Warning: high explicit sexual content.
S lies down on his back and takes W up his hole.
S jacks himself off while W fucks him (W: “love that hole”). S comes on his own belly.
If you’re following the arrangement of bodies here, S could come on W’s belly or on his own; his own belly stipulates the latter, while W’s belly (with a referential full NP W’s) would stipulate the former.
Using referential full NPs (rather than pronouns) is sometimes the only way to keep the participants clear. Here’s another passage, describing a three-person encounter in Manifest (with D as well as S and W):
D lies down on his back, licks W’s hole while S blows D, then S jacks D off.
The two boldfaced occurrences of D are the crucial ones here. The pronoun him for either would be unclear as to whether D or W was getting blown / jacked off; the arrangement of bodies would allow for either.
Just a few simple examples illustrating the choices of referential expressions.