Downloading older games

I recently downloaded "...Ethan Carter" from GOG (I love it, but that's not today's question)and was impressed with the ease of the process. I own a few games which I can't seem to load or run from the disk. Is it possible that a same game, purchased from GOG, will run even if my boxed version won't?

It should, The Haze.
GOG works pretty hard to make sure the games they sell work right out of the box...er, digital download.
Look at the info on the game and what version of Windows or Mac it will run on.
Old games that require DOS have DOS built into them by GOG, btw.
And there's the 30 day money back guarantee they offer if the game gives you fits/won't run and Support can't do anything for you.

Gil.

I've got DOS games from 1995 that I got from GOG running like brand new ones on my Windows 8.1.

What games? How old?
If they're DOS games, or early Windows games that contain 16-bit code, they can't be run on 64bit Windows natively and have to be run inside an emulator like DOSBox. GOG usually puts the DOS games they sell inside DOSBox, though they occasionally use ScummVM instead. They try to make it look like the game is running natively in Windows, but it really isn't.

If you're talking about games made for Windows 98 or XP, GOG may have incorporated patches or tweaks that get rid of some of the problems with running the games on a modern computer. Sometimes it is the copy "protection" that prevents a game from running on a new computer, and GOG supplies their games without any copy "protection" that can block you from playing your legally purchased games.