We discuss the cosmological implications of the new constraints on the power spectrum of the Cosmic Microwave Background Anisotropy derived from a new high resolution analysis of the maxima-i measurement . The power spectrum indicates excess power at \ell \sim 860 over the average level of power at 411 \leq \ell \leq 785. This excess is statistically significant on the \sim 95 % confidence level . Its position coincides with that of the third acoustic peak as predicted by generic inflationary models , selected to fit the first acoustic peak as observed in the data . The height of the excess power agrees with the predictions of a family of inflationary models with cosmological parameters that are fixed to fit the CMB data previously provided by boomerang-ldb and maxima-i experiments . Our results , therefore , lend support for inflationary models and more generally for the dominance of adiabatic coherent perturbations in the structure formation of the Universe . At the same time , they seem to disfavor a large variety of the non-standard ( but inflation-based ) models that have been proposed to improve the quality of fits to the CMB data and consistency with other cosmological observables . Within standard inflationary models , our results combined with the COBE -DMR data give best fit values and 95 % confidence limits for the baryon density , \Omega _ { b } h ^ { 2 } \simeq 0.033 { \pm 0.013 } , and the total density , \Omega = 0.9 { +0.18 \atop - 0.16 } . The primordial spectrum slope ( n _ { s } ) and the optical depth to the last scattering surface ( \tau _ { c } ) are found to be degenerate and to obey the relation n _ { s } \simeq ( 0.99 \pm 0.14 ) +0.46 \tau _ { c } , for \tau _ { c } \leq 0.5 ( all 95 % c.l . ) .