It's very difficult for me to separate them.
I was just listening to some video and it said "Fat cells can’t reproduce themselves." What I thought I've heard is "... CAN reproduce ..."
Frankly, that's pretty annoying.
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The strongest cue for distinguishing these two words is the length of the vowel (in American English at least).
On the telephone, where the auditory signal is compressed, exaggerating the length of the vowels ("did you say ca-an or can't?") is how most people distinguish these two words.
(The cause of this vowel length difference is the presence of the voiceless stop /t/ at the end. Even if the /t/ is realized as the glottal stop (which it often is), this has the same effect.)
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I'm afraid the opposite to what Kosmonaut says applies in the case of British English.
Can rhymes with can (the object), ban, tan, man.
Can't rhymes with car, bar, mar.
In my dialect of American English (mid-western), the unstressed "can" is generally pronounced [ken] or [kən], whereas "can't" is always pronounced with a short "a", as [kænt]. In a stressed position, it's [kæn] vs. [kænt], but the final 't' sound is always aspirated instead of glottal-stopped, making the distinction fairly easy to recognize.