By 1966, Cassels could state “it has been widely conjectured that there is an upper bound for the rank depending only on the groundfield” though he gives no citation and then says “this seems to me implausible” (meaning conjecturing from scanty data, or the rank boundedness itself?) noting that high rank could only occur with large coefficients in any event. At the end of paragraph he states: “We shall give another reason for supposing that the rank is unbounded at the end of S26″, where the finiteness of ш is discussed (I guess he implies that you can get either large rank or large ш, and figures one is no more likely than the other?). Definitely historically, it seems that the 60s were the time when the tipping went to unbounded rank.
See pg257 (pg65 of the PDF) of the survey article “Diophantine Equations with Special Reference to Elliptic Curves” http://jlms.oxfordjournals.org/content/s1-41/1/193.full.pdf

I got this quote from a MathOverflow comment of Milne.

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