An update from 2018: There has been some recent work on a heuristic suggesting that there are infinitely many elliptic curves of every rank $<21$ but only finitely many of rank $> 21$ (it's unclear to me what the model says about the case rank $=21.$)

I'm not sure what the full attribution should be, but one can read about these ideas in Bjorn Poonen's ICM article https://arxiv.org/abs/1711.10112v2 and the references within.

An update from 2018: There has been some recent work on a heuristic suggesting that there are infinitely many elliptic curves of every rank $<21$ but only finitely many of rank $> 21$ (it's unclear to me what the model says about the case rank $=21.$)

I'm not sure what the full attribution should be, but one can read about these ideas in Bjorn Poonen's ICM article https://arxiv.org/abs/1711.10112v2 and the references within.

Update 2024: See the article Park, Jennifer; Poonen, Bjorn; Voight, John; Wood, Melanie Matchett, A heuristic for boundedness of ranks of elliptic curves, J. Eur. Math. Soc. (JEMS) 21, No. 9, 2859-2903 (2019). ZBL1469.11173.