Multi-agent LLM systems can outperform single-agent baselines, yet practitioners still cannot predict which design works for a new task or diagnose why one fails. We argue this gap persists largely because the field lacks a diagnostic framework with measurable primitives and testable predictions. We introduce DIANOIA, a three-channel decomposition of multi-agent reasoning gain into coverage, fidelity, and synthesis, each of which is empirically measurable. From this decomposition, we propose a diagnostic protocol that forms qualitative hypotheses about bottleneck channels for tasks with measurable quality signals. We instantiate the protocol as a multi-agent system whose three components mirror the channels: role-diverse proposers for coverage, execution- or pseudo-verification-grounded review for fidelity, and iterative synthesis. On GSM8K, MBPP, and BFCL-SP, our method outperforms strong multi-agent baselines under matched standard configurations; on MBPP, it matches the strongest baseline with approximately 5× fewer tokens and dominates the measured accuracy–token Pareto frontier. Full-scale MBPP replications on Qwen, DeepSeek, and Gemini consistently improve over the corresponding single-model baseline. A prospective held-out evaluation on VitaBench, whose bottleneck hypothesis was recorded before running DIANOIA, improves pass rate from 29.5% to 45.0%. DIANOIA reframes multi-agent design as channel-aware resource allocation: diagnose the likely bottleneck, then invest tokens accordingly.