Assez usually implies "enough" with respect to some conceived notion to which it is enough for. If you want to say that they just don't have much money then you wouldn't use assez.

Ils n'ont pas assez d'argent. -- They don't have enough money. (They might be millionaires but the yacht just exceeds their budget.I

Ils n'ont pas beaucoup d'argent. -- They don't have much money.
Ils ont peu d'argent. -- They have little money.

Ok, thanks. So assez isn't appropriate here. 

Then I must've heard it in different contexts, because I was actually surprised to learn here that it's meaning is of "enough (relative to a specific target)". 

I saw it in formulations like: "assez bien", "assez facile", "je l'aime assez" where i didn't get the feeling it meant anything like "enough".

assez bien -- good enough
assez facile -- easy enough
je l'aime assez -- I like him (well) enough

That's what these phrases translate to, as you know. Just think of assez meaning "enough" and that's pretty much it. Even in English "enough" implies some sort of imagined threshold. What that threshold is depends on context.

In the context of observing a couple eying a huge yacht "they don't have enough money" is interpreted differently then when seeing someone panhandling.