LLM agents are emerging as a key enabler for autonomous wireless network management. Reliably deploying them, however, demands benchmarks that reflect real engineering risk. Existing wireless benchmarks evaluate single isolated capabilities and treat all errors uniformly, missing both cascaded-chain failures and catastrophic unit confusions (\textit{e.g.}, dB vs.\ dBm). We present \wb{}, the first tolerance-aware, tool-integrated benchmark for LLM-based wireless agents. \wb{} is organized as a three-tier cognitive hierarchy: domain knowledge reasoning (WCHW, 1{,}392 items), intent-driven resource allocation (WCNS, 1{,}000 items), and proactive multi-step decisions under mobility (WCMSA, 1{,}000 items). Moreover, \wb{} is established on three design principles: \emph{(i)}~tolerance-aware scoring with catastrophic-error detection; \emph{(ii)}~tool-necessary tasks requiring a 3GPP-compliant ray-tracing query for channel quality; and \emph{(iii)}~Chain-of-Thought (CoT)-traceable items, where every benchmark item ships with a complete CoT trajectory enabling fine-grained diagnosis of where in the reasoning chain an agent fails. Our numerical results show that the direct-prompting model (GPT-4o) scores $68\%$, trailing a tool-integrated agent ($84.64\%$) by $16.64$\,pp; $23\%$ of errors are catastrophic failures invisible to exact-match metrics. More importantly, the hierarchy decomposes errors into four actionable diagnostic categories that flat evaluation cannot reveal. Code and data: https://wirelessbench.github.io/.