Here's the context. I am guessing it means stumble.
其他选手则要么在微妙的措辞上卡了壳
Other contestants stumble on delicate phrasing
But according to this it means quiet/silenced?
Q: 步枪卡了壳是什么意思？ A: 步枪卡了壳 - 不响 咯咯
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1It means a cartridge got jammed inside the rifle, unable to fire off. Metaphorically, it is used by the author to describe words getting stuck in the mouth. – 杨以轩 Dec 18 '13 at 14:37
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It's a kind of analogy. Originally, 卡壳 (qiǎ ké) meant "a cartridge got jammed inside the gun."
When people are talking very fast and fluently, they are like a gun continuing shooting without any interruption. Words are "shot" (spoken) fast and continuously. There is also an expression "说话像机关枪一样," the literal meaning of which is that someone "speaks like firing a machine gun."
So when someone gets stuck while speaking, we say that he or she "卡壳".
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卡壳 means "getting a stuck cassette" literally. cassette most probably refer to the one holding bullets in a shotgun. But it's also commonly used to say someone getting stuck in the middle of a speech.
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I think this word 卡壳了 can also be used in a situation like the following. (not only referring to speech):
However, for performing the piece, memorizing it HS (=Hand-Separated) will be useful for recovering from blackouts,
（“卡壳了”, or its noun form 卡壳 is a pretty good translation of "blackouts" here）
(From Fundamentals of Piano Practice PDF Page 107, link)
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it means the round is stuck in the barrel or sometimes it means your thought is stuck and can't come up with a solution
