We present a study of the compatibility of some current models of the diffuse Galactic continuum γ-rays with EGRET data. A set of regions sampling the whole sky is chosen to provide a comprehensive range of tests. The range of EGRET data used is extended to 100 GeV. The models are computed with our GALPROP cosmic-ray propagation and γ-ray production code. We confirm that the "conventional model" based on the locally observed electron and nucleon spectra is inadequate, for all sky regions. A conventional model plus hard sources in the inner Galaxy is also inadequate, since this cannot explain the GeV excess away from the Galactic plane. Models with a hard electron injection spectrum are inconsistent with the local spectrum even considering the expected fluctuations; they are also inconsistent with the EGRET data above 10 GeV. We present a new model that fits the spectrum in all sky regions adequately. Secondary antiproton data were used to fix the Galactic average proton spectrum, while the electron spectrum is adjusted using the spectrum of diffuse emission itself. The derived electron and proton spectra are compatible with those measured locally considering fluctuations due to energy losses, propagation, or possibly details of Galactic structure. This model requires a much less dramatic variation in the electron spectrum than models with a hard electron injection spectrum, and moreover it fits the γ-ray spectrum better and to the highest EGRET energies. It gives a good representation of the latitude distribution of the γ-ray emission from the plane to the poles, and of the longitude distribution. We show that secondary positrons and electrons make an essential contribution to Galactic diffuse γ-ray emission.