My radio clock reverts to a quartz timer when the signal goes away, which means it has a ratehr similar failure mode to normal clocks.

I think that's fairly normal, yes; but that's only what happens in the case of a complete and clean loss of signal. If it misreceives the signal, might it not perfectly well suddenly reset itself to a crazy time?

(I found an actual spec once for the MSF signal. I don't recall it having particularly good error correction, although I suppose if it's clear and powerful enough it might not matter. I was terribly tempted to try to build a receiver for it, but I think that's just my standard instinct on seeing a clear specification for anything. :-)