In this paper, we investigate the robustness of structured frames to measurement noise and erasures, with the focus on Gabor frames $(g, Λ)$ with arbitrary sets of time-frequency shifts $Λ$. This property of frames is important in many signal processing applications, from wireless communication to phase retrieval and quantization. We aim to analyze the dependence of the frame bounds of such frames on their structure and cardinality and provide constructions of nearly tight Gabor frames with small $\vert Λ\vert$. We show that the frame bounds of Gabor frames with a random window $g$ and frame set $Λ$ show similar behavior to the frame bounds of random frames with independent entries. Moreover, we study uniform estimates for the frame bounds in the case of measurement erasures. We prove numerical robustness to erasures for mutually unbiased bases frames with erasure rate up to 50\% and show that the Gabor frame generated by the Alltop window can provide results similar to the best previously known deterministic constructions.