I am looking for a simple (if possible) example of a spectral sequence of a filtered complex which does not converge weakly. I don't have great intuition for this, so I'm looking for a simple example. An explicit example would be great.
I think we cannot go with bounded below sequences - is this correct?
It can depend as much on the filtration as on the chain complex. Let $C$ be the chain complex $0 \to k \to 0$, where $k$ is a field. Filter it as follows: for each integer $n$, let $F_n C = C$. Then the associated graded is $F_n/F_{n-1} = 0$ for all $n$, so the spectral sequence is 0, so it certainly does not converge.
Alternatively you could filter by letting $F_n C = 0$ for all $n$ and get just as bad a spectral sequence.