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<p>The Cryogenian period (720–635 million years ago) in the Neoproterozoic era featured two phases of global or near-global ice cover termed “snowball Earth”. Climate models of all kinds indicate that the inception of these phases must have occurred in the course of a self-amplifying ice–albedo feedback that forced the climate from a partially ice-covered to a snowball state within a few years or decades. The maximum concentration of atmospheric carbon dioxide (<span class="inline-formula">CO₂</span>) allowing such a drastic shift depends on the choice of model, the boundary conditions prescribed in the model, and the amount of climatic variability. Many previous studies reported values or ranges for this <span class="inline-formula">CO₂</span> threshold but typically tested only a very few different boundary conditions or excluded variability due to volcanism. Here we present a comprehensive sensitivity study determining the <span class="inline-formula">CO₂</span> thresholds in different scenarios for the Cryogenian continental configuration, orbital geometry, and short-term volcanic cooling effects in a consistent model framework using the climate model of intermediate complexity CLIMBER-3<span class="inline-formula"><i>α</i></span>. The continental configurations comprise two palaeogeographic reconstructions for each of both snowball-Earth onsets as well as two idealised configurations with either uniformly dispersed continents or a single polar supercontinent. Orbital geometries are sampled as multiple different combinations of the parameters obliquity, eccentricity, and argument of perihelion. For volcanic eruptions, we differentiate between single globally homogeneous perturbations, single zonally resolved perturbations, and random sequences of globally homogeneous perturbations with realistic statistics. The <span class="inline-formula">CO₂</span> threshold lies between 10 and 250 ppm for all simulations. While the thresholds for the idealised continental configurations differ by a factor of up to 19, the <span class="inline-formula">CO₂</span> thresholds for the continental reconstructions differ by only 6 %–44 % relative to the lower thresholds. Changes in orbital geometry account for variations in the <span class="inline-formula">CO₂</span> threshold of up to 30 % relative to the lowest threshold. The effects of volcanic perturbations largely depend on the orbital geometry and the corresponding structure of coexisting stable states. A very large peak reduction in net solar radiation of 20 or 30 W m<span class="inline-formula">⁻²</span> can shift the <span class="inline-formula">CO₂</span> threshold by the same order of magnitude as or less than the orbital geometry. Exceptionally large eruptions of up to <span class="inline-formula">−40</span> W m<span class="inline-formula">⁻²</span> shift the threshold by up to 40 % for one orbital configuration. Eruptions near the Equator tend to, but do not always, cause larger shifts than eruptions at high latitudes. The effects of realistic eruption sequences are mostly determined by their largest events. In the presence of particularly intense small-magnitude volcanism, this effect can go beyond the ranges expected from single eruptions.</p>