Flaws in the process of modulation, or encoding of key bits in the quadratures of the electromagnetic light field, can make continuous-variable quantum key distribution systems susceptible to leakage of secret information. Here, we report such a modulation leakage vulnerability in a system that uses an optical in-phase and quadrature modulator to implement a single sideband encoding scheme. The leakage arises from the limited suppression of a quantum-information-carrying sideband during modulation. Based on the results from a proof-of-concept experiment, we theoretically analyze the impact of this vulnerability and show that the leakage reduces the range over which secret key can be obtained, and can even lead to a security breach, if disregarded. We also study the efficacy of additional trusted noise to counter this vulnerability.