To be able to produce highly oriented and strong fibers from polymer solutions, a high elongational rate during the fiber-forming process is necessary. In the air-gap spinning process, a high elongational rate is realized by employing a high draw ratio, the ratio between take-up and extrusion velocity. Air-gap spinning of lignin–cellulose ionic-liquid solutions renders fibers that are promising to use as carbon fiber precursors. To further improve their mechanical properties, the polymer orientation should be maximized. However, achieving high draw ratios is limited by spinning instabilities that occur at high elongational rates. The aim of this experimental study is to understand the link between solution properties and the critical draw ratio during air-gap spinning. A maximum critical draw ratio with respect to temperature is found. Two mechanisms that limit the critical draw ratio are proposed, cohesive breach and draw resonance, the latter identified from high-speed videos. The two mechanisms clearly correlate with different temperature regions. The results from this work are not only of value for future work within the studied system but also for the design of air-gap spinning processes in general.