"Ingen kan veta allt."
Both are fine, but they mean different things.
kunna allt = Have the ability to do everything, be able to do everything.
veta allt = Have knowledge of everything.
It's not as straightforward as that of course, there are nuances that are hard to translate, because the two verbs overlap in ways different from English. For example, kunna generally refers to ability, and veta to knowledge, but you'd use kunna about knowing a skill (how to knit, how to speak French). Indeed, "Do you know this word?" is best translated by "Kan du det här ordet?".
Correct me if I'm wrong, please: so because I can reason out and explain Swedish grammar and vocab, jag vet svenska; but because it takes me effort to actual use Swedish, jag vet bara lite svenska (because I can speak/write/understand Swedish at all, just not very much of it). Is that correct?
With regards to language you say "kan" (ie "att kunna"). So in your case it would be "jag kan svenska" is "I know Swedish" and "jag kan en del svenska" is "I know a bit of Swedish". Instead of "lite"/"a little" we have a tendency to use "en del", which means "a part (of)".
It's better to think of knowing languages as abilities in Swedish. Also, I would never say "jag kan svenska", I would rather put it in the context of understanding, ie "jag förstår svenska".
A couple of examples:
I can speak (in) Swedish - Jag kan prata (på) svenska
I can write in Swedish - Jag kan skriva på svenska
I know Swedish - Jag kan svenska
I know a little Swedish - Jag kan en del svenska / Jag kan svenska till en viss del
I understand Swedish - Jag förstår svenska
I understand a little Swedish - Jag förstår en del svenska
I understand most spoken Swedish but I would not imply that I'm fluent - Jag förstår det mesta ni säger men jag pratar inte Svenska flytande. (contextual translation though)
Does that make sense @Sninguistics?
I'd disagree with the translation of "Do you know this word?", I would say the contextual translation is "Förstår du det här ordet?" eller "Kan du använda det här ordet?"
"Kan du det här ordet?" would be more of a spelling/writing/other-ability-like-context, ie "Can you spell this word?", not "Förstår du innebörden / meningen / användandet av det här ordet?"
If I understand the above discussion correctly, att kunna is about knowing how to do something (in German etwas können). As english does not have a verb with this notion I translated it as Nobody can do everything which - in my opinion - reflects the notion of having the ability better than Nobody can know everything. Though my answer was flagged as wrong... What do you guys think about that?
Your answer flagged as wrong is correct I would say. "Nobody can do everything" would be "Ingen kan göra allt" in Swedish. The sentence "Ingen kan kunna allt" is very hard to translate, the exact sentence cannot be made in English, neither "Nobody can can everything" nor "Nobody know know everything" makes sense, but in practice it means "Nobody can know everything".