The most-likely cause is that the output of the base
station is down. Make sure its antenna isn't dirty (!!!)
or broken, and position the base station where the
handsets can easily "see" it (RF-wise).

Well, I started to hammer out a chiding reply to what I
interpreted as your overly-simplistic suggestions. I was
getting the "out of range" message or other non-operative
condition even when I was standing right next to the base.

But just to humor you, I wiped a layer of dust off the 4"
plastic-covered base antenna and rotated it to an upright
position. I'm sure it's completely unrelated, but my intermittent
issues are currently in remission. Can RF really not punch
through a bit of dust?

If the dust is conductive (and some is), it might conceivably have an
effect -- "It couldn't hurt!" to clean the antenna. But I suspect it was
mostly that the antenna was misadjusted, or possibly loose.

It usually makes sense to start with the obvious. If you're "out of range"
on /two/ (or more) handsets, that suggests low output from the base station,
not defective handsets. And the fact they operate at all -- and well enough
to display an "out of range" message is likewise suggestive.