Pneumonia (PNA): The combination of elevated white blood cell count (WBC), wet cough, and elevated neutrophils is highly suggestive of a bacterial infection, with pneumonia being the most likely diagnosis. The elevated neutrophils indicate a acute bacterial infection, and the wet cough is consistent with the production of purulent sputum often seen in pneumonia.

Acute Bronchitis: Although typically viral, acute bronchitis can sometimes be caused by bacteria, leading to an elevated WBC and neutrophil count. The presence of a wet cough supports this diagnosis.
Chronic Obstructive Pulmonary Disease (COPD) Exacerbation: Patients with COPD can experience exacerbations characterized by increased cough, sputum production, and shortness of breath, along with elevated WBC and neutrophils if there is a bacterial component to the exacerbation.
Influenza: While influenza is primarily a viral infection, secondary bacterial infections can occur, leading to elevated WBC and neutrophil counts. The wet cough can be part of the clinical presentation.

Sepsis: Although less likely, sepsis is a life-threatening condition that can present with elevated WBC and neutrophils. A wet cough could be indicative of a pulmonary source of sepsis, such as pneumonia. Missing this diagnosis could be fatal.
Lung Abscess: A lung abscess can present with similar symptoms, including cough, fever, and elevated WBC count. It is less common but requires prompt diagnosis and treatment to prevent serious complications.
Empyema or Pleural Effusion: These conditions involve fluid accumulation in the pleural space, which can become infected, leading to elevated WBC and neutrophils. They are less common but serious complications of pneumonia or other infections.

Cystic Fibrosis Exacerbation: In patients with cystic fibrosis, an exacerbation can present with increased respiratory symptoms, including a wet cough, and elevated inflammatory markers due to bacterial infection.
Bronchiectasis: This condition involves permanent dilation of parts of the airways and can lead to recurrent infections, presenting with chronic cough, purulent sputum production, and elevated WBC and neutrophils during exacerbations.
Tuberculosis (TB): Although TB typically presents with more chronic symptoms, an acute presentation with elevated WBC and neutrophils can occur, especially in the context of a wet cough and in endemic areas or high-risk populations.