Influenza A viruses (IAV) infect many host species, including humans and pigs. Severe combined immunodeficiency (SCID) is a condition characterized by a deficiency of T, B, and/or natural killer (NK) cells. Animal models of SCID have great value for biomedical research. Here, we evaluated the pathogenesis and the innate immune response to the 2009 H1N1 pandemic IAV (H1N1pdm09) using a recently identified line of naturally occurring SCID pigs deficient in T and B lymphocytes that still have functional NK cells. SCID pigs challenged with H1N1pdm09 showed milder lung pathology compared to the non-SCID heterozygous carrier pigs. Viral titers in the lungs and nasal swabs of challenged SCID pigs were significantly higher than in carrier pigs 7 days postinfection, despite higher levels of IL-1β and IFN-α in the lungs of SCID pigs. The lower levels of pulmonary pathology were associated with the T and B cell absence in response to infection. The higher viral titers, prolonged shedding, and delayed viral clearance indicated that innate immunity was insufficient for controlling IAV in pigs. This recently identified line of SCID pigs provides a valuable model to understand the immune mechanisms associated with influenza protection and recovery in a natural host.