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Stochastic gravitational-wave backgrounds (SGWBs) derive from the superposition of numerous individually unresolved gravitational-wave (GW) signals. Detecting SGWBs provides us with invaluable information about astrophysics, cosmology, and fundamental physics. In this paper, we study SGWBs from binary black-hole (BBH) and binary neutron-star (BNS) coalescences in a network of next-generation ground-based GW observatories (Cosmic Explorer and Einstein Telescope) and determine how well they can be measured; this then limits how well we can observe other subdominant astrophysical and cosmological SGWBs. We simulate all-Universe populations of BBHs and BNSs and calculate the corresponding SGWBs, which consist of a superposition of (i) undetected signals, and (ii) the residual background from imperfect removal of resolved sources. The sum of the two components sets the sensitivity for observing other SGWBs. Our results show that, even with next-generation observatories, the residual background is large and limits the sensitivity to other SGWBs. The main contributions to the residual background arise from uncertainties in inferring the coalescence phase and luminosity distance of the detected signals. Alternative approaches to signal subtraction would need to be explored to minimize the BBH and BNS foreground in order to observe SGWBs from other subdominant astrophysical and cosmological sources.