I have just fished a 4mm plug of pocket lint out from the headphone jack in my iPhone. This lint was preventing full insertion of the headphones. I imagine that pushing down even harder to get them so sit properly just made the lint extraction that much more difficult. I used a needle to dislodge it, and then made the mistake of sucking on the hole like a straw, thus ending up with lint-in-the-mouth.
Note that I am still anti-case for the iPhone — a case wouldn’t have helped anyways since the headphone jack would still have been exposed to pocket lint.