Why am I getting high SWR readings on my handheld antenna analyzer?

We have received numerous reports of high SWR readings with handheld antenna analyzers like MFJ-259 etc. It turns out that these antenna analyzers easily can be overloaded when used to measure multiband wire antenna like EFHW fed with broadband  RF transformer. It is enough to have a Broadcasting station nearby or HF band (1-30MHz) is wide open producing numerous strong HF signals which overload the analyzer. The common cure is to ground the analyzer or simply check your SWR readings with SWR meter at your radio or use external SWR bridge.