@Pmm,
I am a software engineer, not a hardware expert (and I count the wifi router as hardware), but...
I don't buy that explanation at all. Most routers should handle hundreds of connections. The symptom of having too many is simply very slow internet service, not dropped connections and the idea that all of the connections dropping at once because of low bandwidth sounds ridiculous. I think their explanation is nonsense.
It sounds to me like there is a problem with the router. The ISP should provide you with a new one (if it is there).
By the way, you can buy your own router. Many times this will even save you money over the long term.