What does wear thin mean?

[Magomed]

Test No. incompl/advan-37 "A Public Figure", question 3

People tended not to believe a word he said and maintained that his protestations of innocence were wearing a bit ..........

(a) scarce
(b) bare
(c) scant
(d) thin

Test No. incompl/advan-37 "A Public Figure", answer 3

People tended not to believe a word he said and maintained that his protestations of innocence were wearing a bit thin.

Correct answer: (d) thin

Your answer was: incorrect
People tended not to believe a word he said and maintained that his protestations of innocence were wearing a bit bare.
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Why word baer isn't fit here?

[Torsten]

In the given context to wear thin means to become weak. You can wear bare feet or legs, this means, your feet or legs are bare.
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