Have to admit I had no idea previously about weak limits, but thanks to nLab weak limits, maybe we will be able to understand the following example: every non-empty (aka inhabited) set is a weak limit (namely, a weak final object) in the category of sets. But only sets with just one element are final objects; that is, true limits.
Why? Because, given any non-empty set $S$, there is always a map $f: A \longrightarrow S$ from any other set $A$ (pick any element $s\in S$ and send all elements in $A$ to $s$, for instance). But the map $f$ is unique only when $S$ has just one element, obviously.
(Ha! They're funny those weak limits. :-D )