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It doesn't matter if I use the GOG downloader or turn off the GOG downloader to download Deadly Premonition as 3 different files. I can't finish the download.

I have since turned off Sleep mode for my computer since it was interrupting the download. Now the Gog downloader hangs up and won't finish downloading the game.

So, I turned off the GOG downloader and tried downloading DP as three seperate files. I could only finish downloading the first file. The second file keeps hanging up.

I don't know what to do.
It would be in your best interest to contact GOG's Support team. You can do so HERE. Be sure to give them as much information about your system specs and the issue itself as you can. It's always better to give too much information than too little when it comes to tech support. Keep in mind that GOG's Support team is located in Poland, so they may not get back to your right away due to time zone differences, but they will get back to you within a day for sure.

Best of luck!
I am not sure if this will work, but have you tried to erase the currently downloaded files? Maybe they are messed up and avoid, somehow, fully downloading the game. If GOG takes time to answer, I think it won't hurt to try. Just go the the folder, erase everything and clean temp files just in case.