Sarcoidosis: This condition is known for causing enlarged lymph nodes, neuropathy due to granulomatous infiltration of nerves, and respiratory symptoms like cough due to pulmonary involvement. It's a systemic disease that can affect multiple organs, making it a plausible explanation for the combination of symptoms presented.

Lymphoma: Both Hodgkin's and non-Hodgkin's lymphoma can cause enlarged lymph nodes and systemic symptoms. Neuropathy can occur due to direct nerve infiltration or as a paraneoplastic syndrome. Cough could be related to mediastinal involvement.
Tuberculosis (TB): TB can cause lymphadenopathy, neuropathy (especially in the form of radiculopathy or mononeuritis multiplex due to spinal or meningeal involvement), and cough (due to pulmonary TB). It's a consideration, especially in endemic areas or in individuals with risk factors.
HIV/AIDS: Infection with HIV can lead to lymphadenopathy, various forms of neuropathy (due to the virus itself or opportunistic infections), and cough (from opportunistic infections like Pneumocystis jirovecii pneumonia). The immune suppression associated with HIV/AIDS makes patients susceptible to a wide range of pathogens.

Lung Cancer: Although less likely to cause neuropathy directly, lung cancer can lead to cough and lymphadenopathy. Certain types of lung cancer, like small cell carcinoma, can cause paraneoplastic syndromes, including neuropathies. Missing this diagnosis could be fatal due to the aggressive nature of lung cancer.
Wegener's Granulomatosis (Granulomatosis with Polyangiitis, GPA): This is a form of vasculitis that can affect the lungs (causing cough), nerves (leading to neuropathy), and can cause lymphadenopathy. It's a critical diagnosis to consider because of its potential for severe organ damage if not treated promptly.

Castleman Disease: A rare disorder that involves an overgrowth of cells in the body's lymph nodes. It can cause lymphadenopathy, and some forms can lead to systemic symptoms, including neuropathy and potentially cough if there's pulmonary involvement.
Erdheim-Chester Disease: A rare form of histiocytosis that can lead to lymphadenopathy, neuropathy due to infiltration of nerves, and respiratory symptoms including cough. It's a diagnosis that would be considered with a high index of suspicion in the right clinical context.