The influence of multiple vibrational modes on current fluctuations in electron transport through single-molecule junctions is investigated. Our analysis is based on a generic model of a molecular junction, which comprises a single electronic state on the molecular bridge coupled to multiple vibrational modes and fermionic leads, and employs a master equation approach. The results reveal that in molecular junctions with multiple vibrational modes already weak to moderate electronic-vibrational coupling may result in high noise levels, especially at the onset of resonant transport, in accordance with experimental findings of Secker et al..[1] The underlying mechanisms are analyzed in some detail. [1] D. Secker et al., Phys. Rev. Lett. 106, 136807 (2011).