Hello,
I just bought couple of games, and I want to download them using Free Download Manager. I supply correct username and password to this utility (tested, I'm pretty sure), but I'm still getting "Access denied, invalid username or password" error. But I WANT to use FDM for downloading games from GOG.

I have used FDM to download all my games and have had no problems at all (other than the issue with GOG download URLs expiring before you finish downloading so you can't resume, of course, but that has nothing to do with FDM).
If you are using Firefox then I suggest that you install the FlashGot add-on and configure it to use FDM. FlashGot is an firefox add-on which integrates external download managers very nicely into the browser. With this you can simply go to your account page and click on the link to download your game and the download will automagically get handled by your chosen download manager.

The GOG downloader exists for the exact purpose of having a download manager that handles all the authentication and whatnot required for downloads from the site. You could just use that if all else fails.

Yep, I was using DownThemAll! on Firefox to download games before the GOG Team released the Downloader, an that's a whole another story from any standpoint for games download....

Except that the GoG downloader max at 8kb/s for me, so it takes forever to download anything. Is there any reliable way to download GoG's games without the GoG downloader?

Through the browser. There's no reliable way to download anything if your connection isn't reliable in the first place.

I have had no problems using Free Download Manager with GOG, except for the unavoidable problem of links expiring. You do not need to supply a user name and password; as long as your browser is passing the referrer correctly you should be able to simply send the link to FDM and click OK to start the download.
If the source URL stops working, just paste in a new one and resume the download. It couldn't be simpler.

I just downloaded it, and it downloads at 3.5 mb/s which is about 2 megs faster than browser speed.
So fallout 2 in 5 minutes. Not bad.

Ah, but you should still be able to give your username and password to FDM; this would eliminate any worries. I can already do this on my cell phone (!!) using Loader Droid, as LD has a built-in browser. FDM usually works, just sometimes the links expire. It downloads faster on slow / spotty connections because it has the ability to pause / resume as well as downloading multiple segments of the same file at a time. So if one or two segments stall out for a bit, the others pick up the slack.

NECROMONGERS!

Go easy, this is the most excitement I've had today.

I will not convert!