**Core Concept**
Pneumococcal infections, such as meningitis, caused by Streptococcus pneumoniae, can recur in individuals with compromised immunity. The presence of gram-negative diplococci, however, suggests a different pathogen, Neisseria meningitidis, which is known to cause recurrent infections in some patients.

**Why the Correct Answer is Right**
Neisseria meningitidis infections can recur due to the organism's ability to evade the host's immune system. A key factor in this evasion is the complement system, which plays a crucial role in eliminating pathogens. Complement deficiencies, particularly C5-C9, can impair the host's ability to form the membrane attack complex (MAC), leading to recurrent infections with Neisseria meningitidis. This is because the MAC is essential for lysing the bacterial cell membrane, and its deficiency allows the bacteria to survive and cause recurrent infections.

**Why Each Wrong Option is Incorrect**
**Option B:** Immunoglobulin deficiency would affect the humoral immune response, making patients more susceptible to a wide range of infections, not specifically Neisseria meningitidis.
**Option C:** T cell defects primarily affect cell-mediated immunity, which is not the primary mechanism by which Neisseria meningitidis is eliminated.
**Option D:** B cell defects, including immunoglobulin deficiencies, would impair the production of antibodies, but Neisseria meningitidis infections are not typically due to a lack of antibody production.

**Clinical Pearl / High-Yield Fact**
Patients with recurrent Neisseria meningitidis infections should be investigated for complement deficiencies, particularly C5-C9, as these can significantly increase the risk of future infections.

**✓ Correct Answer: A. Complement deficiency**