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We present new upper limits on the neutral hydrogen (H i) content within the stellar half-light ellipses of 15 Galactic dwarf spheroidal galaxies (dSphs), derived from pointed observations with the Green Bank Telescope (GBT) as well as Arecibo L-band Fast ALFA survey and Galactic All-Sky Survey data. All of the limits $ M_ {{\rm H}\,\scriptsize {I}}^{\rm lim} $ are more stringent than previously reported values, and those from the GBT improve upon constraints in the literature by a median factor of 23. Normalizing by V-band luminosity L V and dynamical mass M dyn, we find $ M_ {{\rm H}\,\scriptsize {I}}^{\rm lim}/L_V\sim 10^{-3}\,{M_ {\odot}/L_ {\odot}} $ and $ M_ {{\rm H}\,\scriptsize {I}}^{\rm lim}/M_ {\rm dyn}\sim 5\times 10^{-5} $, irrespective of location in the Galactic halo. Comparing these relative H i contents to those of the Local Group and nearby neighbor dwarfs compiled by McConnachie, we find that the Galactic …