We consider an uncertain linear inverse problem as follows. Given observation $ω=Ax_*+ζ$ where $A\in {\bf R}^{m\times p}$ and $ζ\in {\bf R}^{m}$ is observation noise, we want to recover unknown signal $x_*$, known to belong to a convex set ${\cal X}\subset{\bf R}^{n}$. As opposed to the "standard" setting of such problem, we suppose that the model noise $ζ$ is "corrupted" -- contains an uncertain (deterministic dense or singular) component. Specifically, we assume that $ζ$ decomposes into $ζ=Nν_*+ξ$ where $ξ$ is the random noise and $Nν_*$ is the "adversarial contamination" with known $\cal N\subset {\bf R}^n$ such that $ν_*\in \cal N$ and $N\in {\bf R}^{m\times n}$. We consider two "uncertainty setups" in which $\cal N$ is either a convex bounded set or is the set of sparse vectors (with at most $s$ nonvanishing entries). We analyse the performance of "uncertainty-immunized" polyhedral estimates -- a particular class of nonlinear estimates as introduced in [15, 16] -- and show how "presumably good" estimates of the sort may be constructed in the situation where the signal set is an ellitope (essentially, a symmetric convex set delimited by quadratic surfaces) by means of efficient convex optimization routines.