Slow-roll inflation may simultaneously solve the horizon problem and generate a near scale-free fluctuation spectrum $P(k)$. These two processes are intimately connected via the initiation and duration of the inflationary phase. But a recent study based on the latest {\it Planck} release suggests that $P(k)$ has a hard cutoff, $k_{\rm min}\not=0$, inconsistent with this conventional picture. Here we demonstrate quantitatively that most---perhaps all---slow-roll inflationary models fail to accommodate this minimum cutoff. We show that the small parameter $\epsilon$ must be $\gtrsim 0.9$ throughout the inflationary period to comply with the data, seriously violating the slow-roll approximation. Models with such an $\epsilon$ predict extremely red spectral indices, at odds with the measured value. We also consider extensions to the basic picture (suggested by several earlier workers) by adding a kinetic-dominated or radiation-dominated phase preceding the slow-roll expansion. Our approach differs from previously published treatments principally because we require these modifications to---not only fit the measured fluctuation spectrum, but to simultaneously also---fix the horizon problem. We show, however, that even such measures preclude a joint resolution of the horizon problem and the missing correlations at large angles.