The organization of US presidential elections make them potentially vulnerable to so-called “voting paradoxes”, identified by social choice theorists but rarely documented empirically. The presence of a record high number of candidates in the 2016 Republican Party presidential primaries may have made this possibility particularly latent. Using polling data from the primaries we identify two possible cases: Early in the pre-primary (2015) a cyclical majority may have existed in Republican voters’ preferences between Bush, Cruz and Walker—thereby giving a rare example of the Condorcet Paradox. Furthermore, later polling data (March 2016) suggests that while Trump (who achieved less than 50% of the total Republican primary vote) was the Plurality Winner, he could have been beaten in pairwise contests by at least one other candidate—thereby exhibiting a case of the Borda Paradox. The cases confirm the empirical relevance of the theoretical voting paradoxes and the importance of voting procedures.