Traps limit the photovoltaic efficiency and affect the charge transport of optoelectronic devices based on hybrid lead halide perovskites. Understanding the nature and energy scale of these trap states is therefore crucial for the development and optimization of solar cell and laser technology based on these materials. Here, the low-temperature photoluminescence of formamidinium lead triiodide (HC(NH2)2PbI3) is investigated. A power-law time dependence in the emission intensity and an additional low-energy emission peak that exhibits an anomalous relative Stokes shift are observed. Using a rate-equation model and a Monte Carlo simulation, it is revealed that both phenomena arise from an exponential trap-density tail with characteristic energy scale of ≈3 meV. Charge-carrier recombination from sites deep within the tail is found to cause emission with energy downshifted by up to several tens of meV. Hence, such phenomena may in part be responsible for open-circuit voltage losses commonly observed in these materials. In this high-quality hybrid perovskite, trap states thus predominantly comprise a continuum of energetic levels (associated with disorder) rather than discrete trap energy levels (associated, e.g., with elemental vacancies). Hybrid perovskites may therefore be viewed as classic semiconductors whose band-structure picture is moderated by a modest degree of energetic disorder.