WetGeek So before I give up on this computer I figured I'd ask you whether there's something else I should check. The wi-fi switch on the side causes the wi-fi led on the keyboard to glow, but not the Bluetooth one. I've double-checked that the bios assures me that Bluetooth is controlled by that same switch. Anything else to check?
I suppose you could check the internet, but I can't think of anything off hand. I suppose that it could be a driver issue, but because the light is not working, it sounds more like hardware, specifically something not right with the way the radio interacts with the motherboard - a non-working pin on the motherboard socket or something like that. I'm not even sure where I'd start looking to solve it.
Before you toss the computer (Precision workstations are really good hardware), consider buying a Bluetooth USB adapter. My test computer (Dell Optiplex Micro 5070, refurbished and 5-6 years old now) didn't have an internal Bluetooth (business desktops of the era usually didn't), so I bought a TP-Link UB400 nano adapter, and it works with Linux. Since you are using Bluetooth to attach to a keyboard/mouse, which doesn't take much "bandwidth", an external USB adapter might do the trick, just as "wireless" keyboards and mice do fine with a nano adapter .
Just something to think about, but if you go that route, be careful to check Linux compatibility very carefully (just as you would with an external wifi adapter) or you'll be tossing your money into the ditch.
WetGeek One last question: I now have two new radios for the 4800 - I bought one a while ago to accommodate these Bluetooth peripherals, because the original was one generation too old. So now I have two new radios. If I should get a Latitude next, instead of a Precision, will those likely fit it? (I really like my wife's Latitude.)
Obviously, I don't know, but Dell tends to use relatively standard components in its business computers, so there's a fair chance that it might work.
Dell has an excellent support site, including an "Owner's Manual" that is detailed, so you might be able to figure it out from the support site if you know the model you are considering.
Here's how to get at the "Owners Manual", using a Dell Precision M4800 as an example. Go to the Dell website (dell.com), hover over the "Support" tab to open the drop down menu, and click "Support Home". On the page that shows up, put the model (e.g. "Precision M4800" into the "Identify your product" box, and a support page for that model will open:

Click on "Documentation", and on the page that opens, scroll down to "Manuals and Documents" and you'll find the Owner's manual for the computer (in this case the "Dell Precision Mobile Workstation M4800 Owner's Manual"). The file is available either in pdf or html, and will give you a lot of information about the computer, including tear-down instructions and detailed specifications. I'm thinking that the tear-down instructions might be helpful because you'll likely see photos of the adapter and dock.
If you do that for your Precision and a Latitude (or any other Dell) you are thinking about, comparing the two Owner's manuals should give you a good idea about whether or not your owned radios can be swapped into the new computer or not.
Good luck with it.