From my experience, this usually happens when some other application does some action in the background.
For example, yesterday I played Fallout 1, and at the same time I was downloading something from rapidshare via rapget.The moment rapget finished downloading that file, my screen went black because rapget tried to show some message like "file has been downloaded" or something.
So I quit game, exit rapget and all unnecesarry background applications and I haven't had a black screen ever since.
Actually, the Fallout from GOG works extremely well, without a single crash or bug or anything.
I am using this setup, I wrote it in FAQ at gog.com forums:
The procedure goes as following:
1) Install Fallout 1 from GOG.com
2) Find and install "Fallout_1_TeamX_Patch_ENG_1.2w" (either via nma or just google)
3) Install "f1child_patch" (again, nma)
4) Install "fallup13" - this is the TeamX patch 1.3.4 (nma)
5) Install "f1npcmod" from TeamX (again, nma)
6) Install "sfall1" (from http://timeslip.chorrol.com/sfall.html)
7) Install "Fallout1_High_Resolution_Patch_v1.3" (nma)
8) Open f1_res and set AUTO_SCRN_SIZE = 0, then your desired resolution (I use 800*600, anything above is just too darn small for my taste), then if you want, change the way ingame movies work, either fullscreen or not....up to you.Same with movie subtitles.
Run the game and enjoy it in all of its patched glory, WITH KILLABLE KIDS (not that I ever kill them, but the option is here), with better NPC interaction (armors, levelling..), with tweaks like mousewheel in the inventory etc, in higher resolution...