Integrated spectroscopy is the method of choice for deriving the ages of unresolved stellar systems. However, hot stellar evolutionary stages, such as hot horizontal branch stars and blue straggler stars (BSSs), can affect the integrated ages measured using Balmer lines. Such hot, ``noncanonical'' stars may lead to overestimation of the temperature of the main-sequence turnoff, and therefore underestimation of the integrated age of a stellar population. Using an optimized Hβ index in conjunction with HST WFPC2 color-magnitude diagrams (CMDs), we show that Galactic globular clusters exhibit a large scatter in their apparent ``spectroscopic'' ages, which does not correspond to that in their CMD-derived ages. We find for the first time that the specific frequency of BSSs, defined within the same aperture as the integrated spectra, shows a clear correspondence with Hβ in the sense that, at fixed metallicity, higher BSS ratios lead to younger apparent spectroscopic ages. Thus, the specific frequency of BSSs in globular clusters sets a fundamental limit on the accuracy with which spectroscopic ages can be determined for globular clusters, and perhaps for other stellar systems such as galaxies. The observational implications of this result are discussed.