Thanks for the link!
When I plug in my first USB Drive, the utility indicates that it is connected, safe to remove, and consuming 100mA.
As soon as I plug in my second USB stick; the first one remains connected, but is no longer safe to remove, nor is it consuming any power. And as before, it reports no data when opening the drive in "my computer".
It can't be a power problem since the second drive I just used to test it has its own external power plug.
Any help would be appreciated since the trouble with making backups has already led to loss of data since my last external USB drive gave up on me.