The boundedness of the rank of twists of a fixed curve

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.

This is not a complete answer to your question (since I think it isn't known), but you might still find this interesting.

Mestre has shown in Rang de courbes elliptiques d'invariant donné (http://arxiv.org/abs/alg-geom/9206007) that any elliptic curve over $\mathbf{Q}$ has a twist with rank at least $2$. For $j=1728$, he improves this to rank at least $4$, and for $j=0$ to rank at least $6$. I think he even gets infinitely many such twists in each case, but I'm not sure how easily this follows from what he says (see below though).

Stewart and Top have improved on Mestre's result for arbitrary elliptic curves, by giving a lower bound of the form $cT^a/(\log T)^b$ on the number of quadratic twists with rank at least $2$ and with "twist parameter" below $T$. See Theorem 3 in their On ranks of twists of elliptic curves and power-free values of binary forms (http://www.math.rug.nl/~top/StewartTop.pdf). For the $j=0$ family $y^2 = x^3 + d$, they get similar lower bounds for the number of members with rank at least $2, 3, 4, 5$ and $6$ (Theorem 9).

This doesn't answer your question, but I'll mention it anyway. My thesis research shows that the average rank of $j = 0$ elliptic curves, when ordered by discriminant, is bounded by 1.5. This follows from computing the average size of the 2-Selmer group, which is 3.

I'll also mention that your question is much easier to answer if you replace Mordell-Weil rank with 2-Selmer rank. In that case, Heath-Brown has two papers (The Size of the Selmer Group for the Congruent Number problem I, II) showing that for any $r$, a positive density of the curves $y^2 = x^3-D^2x$ have 2-Selmer rank $r$. Mazur and Rubin have a paper (Ranks of Twists of Elliptic Curves and Hilbert's Tenth Problem) where they consider quadratic twists of curves with no rational 2-torsion and give some conditions for the family of twists to have arbitrary 2-Selmer rank.