I noticed that built-in downloader for Firefox doesn't work well with GOG servers. If I pause the download, I cannot resume. If there is an error, I cannot resume either. Don't your servers support range requests and partial responses?

If you want to pause and resume the download, try using the GOG downloader (install Adobe AIR first). Unless you despise, or cannot run, AIR for some reason.

You won't be able to use conventional means to pause and resume downloads as GOG uses authorization methods that time out, meaning that when you resume a download, you'll have the wrong authorization and your download will not continue. The only way at current to pause and resume downloads is to use the GOG downloaded as Miaghstir suggests.

I'd rater not install Air. I try to keep my gaming system as clean as possible. All the bloat goes into a virtual machine under Virtual Box. Hm... Come to think of it, I can install Air on that system, download games, and then copy to non-virtual disk via shared folders.
Still, it's strange to hear that the authorization immediately times out. Being able to pause and resume via Mozilla would be a good feature to see.

A custom downloader - however great the app itself might be - is a functional but extremely awkward solution. Is there any reason that GOG doesn't use a more conventional approach such as HTTP "Authorization" requests?

The GOG downloader is not an installer, its just a download manager specifically designed to interact with GOG alone. Frankly, I think the only reason they don't do anything else is because they always intended/expected that the GOG downloader would become the default for everyone who uses GOG.

You don't quite sound like the gambling type of man..

Oops - I meant to say "downloader" :P

To do what you are wanting you will need a third-party download manager; Firefox isn't sufficient. I use Free Download Manager Lite. Because GOG's download servers use special authentication every time you want to resume you will need to replace the source URL (via the "download via browser" link); once you do this the download will pick up where it left off as if nothing had happened. If Free Download Manager tries to capture the link and start a new download for it simply stop the second one, copy its URL, and use it for the one you had already started on (alternately, you can disable automatic URL detection altogether in the program settings).

There's probably an add-on that adds that feature.

Firefox will not resume protected downloads (i.e. those that need a password or an account login). It's a security feature. Keeps people from using a resume URL to distribute protected content.
If it wasn't that way, you could start a download, cancel it, send someone else the partial file and the URL, and they could download the rest of the file. It could also allow someone to do a "man in the middle" attack and intercept downloads. With GOG it's probably not that critical since the stuff is DRM free anyway, but that's the way Firefox was designed.
There's probably nothing you can do about it without using the GOG downloader.

All the more critical, I'd say. With DRM'd content you would probably not be able to use it even if you somehow managed to download it. With GOGs games anyone can use the downloaded file.

I use wget on my home server so that it downloads in the background; it's a shame that finding the URL to download is a bit of a pain (I have to start the download in Firefox and then copy the URL to the server), but better than having the wireless network go down when I'm 1GB into a 1.1GB download to my laptop so I have to restart it.