Unfortunately this is not the case with many Geiger counters, especially older ones. Geiger-Muller (GM) tubes detect radiation by measuring ionization events. When a high energy particle passes through the tube, it leaves a little trail through which electricity can flow, but only for a moment — this temporary flow of electricity is amplified and fed into the detection circuit, and it also what is used to generate the typical clicking noise. The Geiger counter is basically an edge detector + summing circuit with decay.
When a GM tube is overloaded, the particles create too many overlapping trails, and the tube cannot stop conducting electricity. This continuous flow of electricity does not trigger the edge detector, so it thinks the level of radiation is zero. Some modern equipment has an alarm for this condition.
