Study of Cord Blood Derived Mesenchymal Stem Cells for Treatment of Moderate, Severe or Critical Pneumonia

PRIMARY OBJECTIVE: 1\. To assess the safety of administering cord blood derived mesenchymal stem cell (CB-MSC) infusions for treatment of moderate, severe or critical pneumonia. SECONDARY OBJECTIVES: 1. In the group of participants who present intubated on ventilator support, assess the proportion that are able to be successfully extubated. 2. In the group of participants who present requiring supplemental oxygen but otherwise breathing without assistance, assess the rate of progression to intubation. 3. Estimate the survival rate at day 30 post treatment separately by group. 4. Determine the treatment effect on clinical parameters, oxygenation and respiratory parameters: * Resolution of fever * Changes in oxygen demand (increased oxygen saturation at similar FiO2 or decreased FiO2 requirement) * Progression to mechanical ventilation * Length of Mechanical ventilation * Decrease in PEEP in intubated participants * Decrease in FiO2 in intubated participants 5. Determine the treatment effect on laboratory markers: * Complete blood count * CRP * Ferritin * D Dimer * Procalcitonin * Cytokine levels 6. Estimate hospitalization and ICU stay. 7. Report on study related adverse events. OUTLINE: Currently not shipping cells outside of MD Anderson Cancer Center in Houston. PILOT STUDY: Patients receive MSCs intravenously (IV) over 1-2 hours on day 1. participants may receive a second infusion of MSCs within 7 days after the first infusion per physician discretion. PHASE II STUDY: Patients are randomized to 1 of 2 arms. ARM I: Patients receive MSCs IV over 1-2 hours on day 1. Patients may receive a second infusion of MSCs within 7 days after the first infusion per physician discretion. ARM II: Patients receive standard of care. After completion of study treatment, patients are followed up at days 7, 14, 30, 60, and months 6 and 12.