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
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".
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.
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)
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
