Integral field spectroscopy (IFS) studies based on CALIFA survey data have recently revealed ongoing low-level star formation (SF) in the periphery of a small fraction (∼10%) of local early-type galaxies (ETGs), witnessing a still ongoing inside-out galaxy growth process. A distinctive property of the nebular component in these ETGs, classified i+, is a structure with two radial zones, the inner of which displays LINER emission with a Hα equivalent width EW(Hα) ≃ 1 Å, the outer (3 Å <EW(Hα) ≲ 20 Å) HII-region characteristics. Using CALIFA IFS data, we empirically demonstrate that the confinement of nebular emission to the galaxy periphery leads to a strong aperture (or, correspondingly, redshift) bias in spectroscopic single-fiber studies of type i+ ETGs: at low redshift (z ≲ 0.45), SDSS spectroscopy is restricted to the inner (SF-devoid LINER) zone, which causes the galaxies to be erroneously classified as "retired", that is, systems entirely lacking SF, and whose faint nebular emissionis solely powered by the post-AGB stellar component. The SDSS aperture progressively encompasses the outer SF zone only at higher z, at which the galaxies are unambiguously classified as "composite SF/LINER". We also empirically demonstrate that the principal effect of a decreasing spectroscopic aperture on the classification of i+ ETGs through standard [NII]/Hα vs. [OIII]/Hβ emission-line (BPT) ratios consists of a monotonic shift upward and to the right precisely along the upper right wing of the "seagull" distribution on the BPT plane, that is, along the pathway connecting composite SF/HII galaxies with AGN/LINERs. Motivated by these observational insights, we also investigate theoretically observational biases in aperture-limited studies of inside-out growing galaxies as a function of z. To this end, we devise a simple 1D model that involves an outward-propagating exponentially decreasing SF process since z ∼ 10 and reproduces the radial extent and two-zone EW(Hα) distribution of local i+ ETGs. By simulating the 3″ spectroscopic SDSS aperture in this model, we find that SDSS studies at z ≲ 1 are progressively restricted to the inner (SF-devoid LINER) zone and miss an increasingly larger portion of the Hα-emitting periphery. This leads to the incorrect spectroscopic classification of these inside-out assembling galaxies as retired ETG/LINERs and also to a severe underestimation of their total star formation rate (SFR) in a manner inversely related to z. More specifically, the SFR inferred from the Hα luminosity registered within the SDSS fiber is reduced by 50% at z ∼ 0.86, reaching only 0.1% of its integral value at z = 0.1. We argue that the aperture-driven biases described above pertain to any morphological analog of i+ ETGs (e.g., SF-quiescent bulges within star-forming disks), regardless of whether it is viewed from the perspective of inside-out growth or inside-out SF quenching, and might be of considerable relevance to galaxy taxonomy and studies of the cosmic SFR density as a function of z.