Connectomes with high sensitivity and high specificity are unattainable with current axonal fiber reconstruction methods, particularly at the macro-scale afforded by magnetic resonance imaging. Tensor-guided deterministic tractography yields sparse connectomes that are incomplete and contain false negatives (FNs), whereas probabilistic methods steered by crossing-fiber models yield dense connectomes, often with low specificity due to false positives (FPs). Densely reconstructed probabilistic connectomes are typically thresholded to improve specificity at the cost of a reduction in sensitivity. What is the optimal tradeoff between connectome sensitivity and specificity? We show empirically and theoretically that specificity is paramount. Our evaluations of the impact of FPs and FNs on empirical connectomes indicate that specificity is at least twice as important as sensitivity when estimating key properties of brain networks, including topological measures of network clustering, network efficiency and network modularity. Our asymptotic analysis of small-world networks with idealized modular structure reveals that as the number of nodes grows, specificity becomes exactly twice as important as sensitivity to the estimation of the clustering coefficient. For the estimation of network efficiency, the relative importance of specificity grows linearly with the number of nodes. The greater importance of specificity is due to FPs occurring more prevalently between network modules rather than within them. These spurious inter-modular connections have a dramatic impact on network topology. We argue that efforts to maximize the sensitivity of connectome reconstruction should be realigned with the need to map brain networks with high specificity.