Regarding the work of Watkins as mentioned by Matt Young, he has a different paper where the question of rank distribution in quadratic twist families is considered. There the RMT prediction for rank 2 is well shown by the Rubinstein data to demonstrate about $D^{3/4}$ twists of rank 2 (or more) in the even parity subclass, while Watkins suggests that 3/4 is too high for rank 3. He does not posit anything exactly, but the data for the congruent number curve, divided into 2 natural classes (Section 3.3) gives best-fit exponents of 0.44 and 0.55, so about $D^{1/2}$ is probably a best current guess for rank 3, though in initial regions of data collection the logarithmic factors can be difficult.

http://archive.numdam.org/article/JTNB_2008__20_3_829_0.pdf

The natural linear extrapolation for these types of problems might be semi-valid at least initially, so $D^{1/4}$ for rank 4 and then some power of logarithm for rank 5 are as knowledgeable as guesses as any. But the 2-torsion plays a role here, and it is not clear whether it affects the exponent on the $D$-power. Watkins has a recent preprint (joint with 5 others) where data for the congruent number curve is given, finding "lots" of rank 6 examples, but "few" of rank 7 (and none of rank 8). Granville has a heuristic (see Section 4 loc. cit.) that might suggest ranks 5 and 7 are the correct cutoffs in the generic and 2-torsion cases.

http://magma.maths.usyd.edu.au/~watkins/papers/RANK7.pdf

But really no one has any factual idea, and a number of caveats can be listed, concerning specially parametrised (sparse) families. Indeed, the rank 28 example of Elkies starts from a rank 17 special family and he gains 11 from searching on specializations, and the same was approximately true for the NSA curve, they had rank 24 starting from I think a rank 13 family of Mestre or Nagao. So again 11 more than the family rank. However, generically it seems one should not expect more than 10 "small" points (polynomial height) on an elliptic curve except in such special families, and one actually reachieves the bound of 21 when appending 11 as above.

Edit: Elkies says that the NSA searched in a rank 11 family, so they beat the family rank by 13 in fact, see page 5 of his arxiv.org/pdf/0709.2908v1