The quest for a half-quantized thermal Hall effect of a Kitaev system represents an important tool to probe topological edge currents of emergent Majorana fermions. Pertinent experimental findings for α-RuCl3 are, however, strongly debated, and it has been argued that the thermal Hall signal stems from phonons or magnons rather than from Majorana fermions. Here, we investigate the thermal Hall effect of the Kitaev candidate material Na2Co2TeO6, and we show that the measured signal emerges from at least two components, phonons and magnetic excitations. This dichotomy results from our discovery that the longitudinal and transversal heat conductivities share clear phononic signatures, while the transversal signal changes sign upon entering the low-temperature, magnetically ordered phase. Our results demonstrate that uncovering a genuinely quantized magnetic thermal Hall effect in Kitaev topological quantum spin liquids such as α-RuCl3 and Na2Co2TeO6 requires disentangling phonon vs magnetic contributions, including potentially fractionalized excitations such as the expected Majorana fermions.