It has been proposed that the globular cluster-like system Terzan 5 is the surviving remnant of a primordial building block of the Milky Way bulge, mainly due to the age/metallicity spread and the distribution of its stars in the α-Fe plane. We employ Sloan Digital Sky Survey data from the Apache Point Observatory Galactic Evolution Experiment to test this hypothesis. Adopting a random sampling technique, we contrast the abundances of 10 elements in Terzan 5 stars with those of their bulge field counterparts with comparable atmospheric parameters, finding that they differ at statistically significant levels. Abundances between the two groups differ by more than 1σ in Ca, Mn, C, O, and Al, and more than 2σ in Si and Mg. Terzan 5 stars have lower [α/Fe] and higher [Mn/Fe] than their bulge counterparts. Given those differences, we conclude that Terzan 5 is not the remnant of a major building block of the bulge. We also estimate the stellar mass of the Terzan 5 progenitor based on predictions by the Evolution and Assembly of GaLaxies and their Environments suite of cosmological numerical simulations, concluding that it may have been as low as ~3 × 108 M⊙ so that it was likely unable to significantly influence the mean chemistry of the bulge/inner disc, which is significantly more massive (~1010 M⊙). We briefly discuss existing scenarios for the nature of Terzan 5 and propose an observational test that may help elucidate its origin.