Hi Al,
Yes, that's possible, as long as you have control over the transmitting devices or if you don't care if you miss some data by cycling thru the devices listening to one device at a time. If they are self contained units sending data on their own, at will, then it's going to be tricky. If I've missed your point please enlighten me :-)

Now, the OP doesn't say how the transmitting devices works and/or if he has "control" over how they works. Hopefully a multidrop network can be implemented in which case one USART would be enough.

/Henrik.