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.
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.)
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.
I'm a native speaker of American English with Chicago/Ohio influences from my parents and have grown up mostly in the South. Here's how I do it:
I "kin" [can] go tomorrow, or, I "kən" [can] go tomorrow. These two vowel options are not necessary perceived as a binary choice, in other words, they easily blend into a hybrid vowel which is mostly subsumed by the nasal n sound. There is a moderately audible exhalation of air on the K, and once the jaws pulls up from the aɪ of the preceding word, the jaw stays in place. There is practically no vowel sound other than the humming of the n. It's really "kn".
Remember, in English we tend to really "hang out" on (extend) the m and n sounds, it's not like Spanish where the tongue or lips meet precisely, separate, and go on to another open vowel. Especially when the m and n sounds come at the end of words, they are treated practically like closed vowels in English.
Also the prosody of the affirmative phrase would tend to include subtle differences in pitch and emphasis. You'd hear some encouraging uptones, emphasizing "I" with a longer, upswinging tone, say, if one is volunteering oneself from among other people. Or if "tomorrow" is the operative assertion (as opposed to today, or next week) then "mər" of "morrow" is emphasized with a slightly higher (more friendly) pitch. These subtle changes in pitch and the lengthening of certain syllables call attention to the positive intent of the phrase.
I "kæ(n)ʔ" [can't] go tomorrow. There is a more audible exhalation with the K sound. The n sound is more a nasalization of the æ, which vowel also requires a movement of the jaw. In terms of prosody, the "I" is quick, and there is a slight rhythmic emphasis on the syllables "can't", "go" and possibly the "ma:" of "morrow". To keep the pitch quick and low on "morrow", notice the vowel may open out from ər of the previous sentence to a:.
In addition to pronunciation differences, there are usually important clues in the surrounding choice of words, and facial expressions, telling you whether this is an affirmation or a negation.
I can't go tomorrow. [brow furrowed, squinty eye contact with listener]
I can't go tomorrow, but maybe [suggests an alternative].
I can't. [point blank refusal. All the earlier pronunciation qualities apply]
They can't go tomorrow because [explanation given with refusal].
I'm sorry, they can't go. [apology gives it away]
Who told you that? I can't go tomorrow, I have to be at work by 8. [heated refusal with reasons]
Or, [all the following are said with a relaxed, open facial expression-- lifted or neutral eyebrows, and a relaxed jaw and lips]
I can. [upbeat prosody]
I can hardly wait! [expression of enthusiasm]
I can go tomorrow if you want. [makes it a conditional offer]
They can go after they've finished eating. [another conditional affirmation]
These are all personal conversations. In other contexts such as the scientific commentary you reference, there should be other clues:
Fat cells can't reproduce themselves. Fat cells can reproduce themselves.
With the second, positive affirmation, you are either going to hear the humming of the n sound of "can" for longer, with a: for the vowel, if the idea is being emphasized. Or the word will collapse down as I explained earlier, into "kn".
With the negative assertion you will hear more air with the K of "can't". You'll hear the nasally æ(n) sound with a glottal stop at the end.
There will probably be clues in the surrounding comments, as well.
Hope that helps.