Aim. The lack of large-angle correlations in the fluctuations of the cosmic microwave background (CMB) conflicts with predictions of slow-roll inflation. But while probabilities (≲0.24%) for the missing correlations disfavour the conventional picture at ≳3σ, factors not associated with the model itself may be contributing to the tension. Here we aim to show that the absence of large-angle correlations is best explained with the introduction of a non-zero minimum wave number kmin for the fluctuation power spectrum P(k). Methods: We assumed that quantum fluctuations were generated in the early Universe with a well-defined power spectrum P(k), although with a cut-off kmin ≠ 0. We then re-calculated the angular correlation function of the CMB and compared it with Planck observations. Results: The Planck 2013 data rule out a zero kmin at a confidence level exceeding 8σ. Whereas purely slow-roll inflation would have stretched all fluctuations beyond the horizon, producing a P(k) with kmin = 0 - and therefore strong correlations at all angles - a kmin ≠ 0 would signal the presence of a maximum wavelength at the time (tdec) of decoupling. This argues against the basic inflationary paradigm, and perhaps even suggests non-inflationary alternatives, for the origin and growth of perturbations in the early Universe. In at least one competing cosmology, the Rh = ct universe, the inferred kmin corresponds to the gravitational radius at tdec.