(All examples from a mailing list post by Henrik Theiling.)
Compare these two phrases:
mit den Jungen
ohne den Jungen
In both cases, you have den Jungen -- but in the first phrase, it unambiguously means "the boys" (plural), and in the second, "the boy" (singular), due to the fact that "mit" takes the dative case (so "den" must be dative plural) but "ohne" takes the accusative case (so "den" must be masculine accusative singular). And this, even though the meanings are very closely related: "with" and "without"!
I immediately understand which is which, but I imagine that for a language learner, this would be more difficult, since you have to internalise which prepositions govern which case to disambiguate such homophonous articles.
Or, for a slightly more contrived example involving not only homophonous articles but words which are distinguished by gender:
Der Finne entspricht der Norm.
Der Finne entspricht der Schwanz.
The first sentence means "The Finn conforms to the norm" and the second means "The tail corresponds to the back fin".
So since "Norm" is feminine singular, "der" here must be feminine genitive or dative singular; in this case, dative, as the object of "entsprechen", so "der Finne" must be masculine nominative singular and mean "the Finn (person from Finland).
In the second sentence, "Schwanz" is masculine singular, so "der" here must be masculine nominative singular and must be the subject of "entsprechen". This means that "der Finne" must be feminine dative singular, the object of "entsprechen", and correspond to the nominative "die Finne", meaning "the (back) fin (e.g. of a fish)".
It's all quite simple really. *snerk*