The fully parabolic cross-diffusion system{εut=Δu−χ∇⋅(uv∇v),vt=Δv−v+u, is considered under homogneous Neumann boundary conditions in a ball Ω⊂Rn, n≥1, for ε∈(0,1) and χ>0. Previous literature has asserted global existence of certain generalized solutions whenever χ<n(n−2)+, but no result on any unboundedness phenomenon seems available; only in a parabolic-elliptic simplification, some solutions are known to blow up in finite time when n≥3 and χ>2nn−2.The present study examines the solution behavior in the singular limit ε↘0, and based on an essentially well-known result on finite-time blow-up in an accordingly obtained nonlocal scalar parabolic problem, under the assumptions that n≥3 and χ>nn−2 a statement on spontaneous emergence of arbitrarily large values of u for appropriately small ε is derived.