Firefox Download Manager – Classic Download Manager for Firefox >20

Try finding network.http.max-connections-per-server in about:config and reducing the value.

The download managers don't require the server to support resuming

That's not correct. If a server doesn't support resume then download managers cannot resume downloads.

When is Firefox going to finally add the ability to resume downloads

Firefox does resume downloads. The problem is that many servers refuse to resume downloads even if there's a break in connection - the time period differs from server to server. ( TED's servers immediately comes to my mind, if there's break in connection for about 5 seconds, the download is ruined and I have to restart the download from start.)

I just tried Free Download Manager (FDM), again, having the Firefox addon Flashgot use it. The download gets passed to FDM, and fails, giving the error message "access denied, invalid username or password.

I also tried DTA instead of FMD with Flashgot, and I get an "access denied" message in DTA.

Few servers inspect the HTTP headers for the referrers, and block if the referrer is not a browser. I guess while transferring the download, the header gets mangled and the referrer becomes the download manager, instead of the browser.

The failure message in the Firefox Downloads window is source file at http... could not be read.

That's become the download links are autogenerated, are valid particular combination of parameters ( perhaps some session id ? I'm not sure), and are not persistent. Once the control of the download has been handed over to the browser, the links expire, and hence you get the error message. My suggestion: Try IDM. It's one of the better download managers that I've used. If the problem still persists, my suggestion is to use an alternative link to download, or maybe switch to a wired connection while downloading.