Changes in resting-state functional connectivity (rs-FC) under general anesthesia have been widely studied with the goal of identifying neural signatures of consciousness. This work has commonly revealed an apparent fragmentation of whole-brain network structure during unconsciousness, which has been interpreted as reflecting a break-down in connectivity and a disruption in the brain's ability to integrate information. Here we show, by studying rs-FC under varying depths of isoflurane-induced anesthesia in nonhuman primates, that this apparent fragmentation, rather than reflecting an actual change in network structure, can be simply explained as the result of a global reduction in FC. Specifically, by comparing the actual FC data to surrogate data sets that we derived to test competing hypotheses of how FC changes as a function of dose, we found that increases in whole-brain modularity and the number of network communities -- considered hallmarks of fragmentation -- are artifacts of constructing FC networks by thresholding based on correlation magnitude. Taken together, our findings suggest that deepening levels of unconsciousness are instead associated with the increasingly muted expression of functional networks, an observation that constrains current interpretations as to how anesthesia-induced FC changes map onto existing neurobiological theories of consciousness.