This is a very interesting question, and there was renewed interest in it lately. As Chris Wuthrich says in the comments, the situation used to be that most people believed in unboundedness of ranks for elliptic curves with a given $j$-invariant, although there was no particular evidence in favour of this. In fact, there was an old conjecture of Honda

T. Honda, Isogenies, rational points and section points of group varieties, Japan. J. Math.,
30 (1960), 84-101

that asserted boundedness, but people did not really believe it. However, recently, there was work by Andrew Granville, Mark Watkins and others with some serious heuristics pointing towards boundedness as well. Mark's slides for his recent talk at the Warwick conference about this are here. In particular, the heuristics suggest that for the family
$$
X^3+Y^3=A\qquad (j=0)
$$
the correct answer could be that the ranks are bounded by $9$, except for finitely many curves in the family. Mark also notes, though, that all these heuristics are quite shaky, and are also very difficult to verify, so at the moment it is really not clear what to expect.

Finally, people working in random matrix theory, notably Nina Snaith, Jon Keating and their collaborators are also looking into higher ranks in the family of quadratic twists. Nina spoke about it here just two weeks ago, and these are her slides. They are working on a precise heuristic for the number of rank 2 twists, with a hope that the approach would eventually lead to a conjectural formula for the number of higher rank twists as well.

In any case, the current status is that the precise answer is not known for a single $j$-invariant.