This paper introduces a new consistency metric, Network Imprecision (NI), to address a central challenge in largescale monitoring systems: safeguarding correctness despite node and network failures. To implement NI, an overlay that monitors a set of attributes also monitors its own state so that queries return not only attribute values but also information about the stability of the overlay—the number of nodes whose recent updates may be missing and the number of nodes whose inputs may be double counted due to overlay reconfigurations. When NI indicates that the network is stable, query results reflect the true state of the system, but when the network is unstable, NI puts applications on notice that query results should not be trusted, allowing them to take corrective action such as filtering out inconsistent results. To implement NI’s introspection scalably, our prototype introduces a key optimization, dual-tree prefix aggregation, which exploits overlay symmetry to reduce overheads by more than an order of magnitude. Evaluation of three monitoring applications demonstrates that NI flags inaccurate results while incurring low overheads, and monitoring applications that use NI to select good information can reduce their inaccuracy by nearly a factor of five.