In this paper we document for the first time some of the effects of self-healing, a property of orbital-angular-momentum (OAM) or vortex beams, as observed on a millimeter-wave experimental communications link in an outdoors line-of-sight (LOS) scenario. The OAM beams have a helical phase and polarization structure and have conical amplitude shape in the far field. The Poynting vectors of the OAM beams also possess helical structures, orthogonal to the corresponding helical phase-fronts. Due to such non-planar structure in the direction orthogonal to the beam axis, OAM beams are a subset of structured light beams. Such structured beams are known to possess self-healing properties when partially obstructed along their propagation axis, especially in their near fields, resulting in partial reconstruction of their structures at larger distances along their beam axis. Various theoretical rationales have been proposed to explain, model and experimentally verify the self-healing physical effects in structured optical beams, using various types of obstructions and experimental techniques. Based on these models, we hypothesize that any self-healing observed will be greater as the OAM order increases. Here we observe the self-healing effects for the first time in structured OAM radio beams, in terms of communication signals and channel parameters rather than beam structures. We capture the effects of partial near-field obstructions of OAM beams of different orders on the communications signals and provide a physical rationale to substantiate that the self-healing effect was observed to increase with the order of OAM, agreeing with our hypothesis.