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 alpha-Fe plane. We employ Sloan Digital Sky Survey (SDSS-IV) data from the Apache Point Observatory Galactic Evolution Experiment (APOGEE 2) 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 1-sigma in Ca, Mn, C, O, and Al, and more than 2-sigma in Si and Mg. Terzan 5 stars have lower [alpha/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 (EAGLE) suite of cosmological numerical simulations, concluding that it may have been as low as ~3x10^8 M_sun so that it was likely unable to significantly influence the mean chemistry of the bulge/inner disk, which is significantly more massive (~10^10 M_sun). We briefly discuss existing scenarios for the nature of Terzan 5 and propose an observational test that may help elucidate its origin.