That's not a bold claim, that's a very safe claim. Learning C won't hurt anyone, except for the missed opportunities, which are invisible. But if you take all the "everybody should learn X" claims literally, you'll spend all your time learning.
So here's my counter-claim. Most Java and .NET developers will do just fine without knowledge of C.
Until you need to do something novel or unique. how useful would .NET be without pinvoke ? how many Java extensions exists out there ?
Above and beyond the practicality of being able to extend your language or debug it when it brakes in new and interesting ways, you also just have the general knowledge of how computers work that much more. This isn't to say you aren't any good, but if you want to be that much better you should learn C.
Server-side and in command-line tools there is tons of software that need not explicitly call into the lower level stuff. The code you call will eventually do that, but as far as the programmer is concerned, that is of no importance.
GUIs may still be a different matter, but I haven't programmed them in .NET/Java in years.
So here's my counter-claim. Most Java and .NET developers will do just fine without knowledge of C.