Your downloader does not come with any means of validating the download. I have had maybe one in four downloads succeed without needing to delete the downloaded file[s] and start it again.
Right now? I'm on my cell phone because I'm not at home. Which means that I've wasted 20% of my bandwidth downloading a game that did not download and that it is impossible to fix save by deleting the installer and starting it again.
Either build a better installer or provide us with validator tools.

Get rid of popups. Seriously, it's 2015. There is absolutely no need for a popup for the forums, they are a pain in the rear and should never be required for basic funtionality. Pop ups are a blight on the face of the internet.

It's strange that your downloads fail. I own more than a hundred GOG games and the downloader never failed me once. Maybe there's a problem with your Internet connection?

On the installer's "options" there's an "Integrity Check", I think. The installer checks whether it's broken before installing.

Also, the pop ups really should maybe just open in a new tab.

Add the download again to the downloader, without moving/deleting the files first. While the downloader will display "Downloading", it is actually hash checking, which is only obvious due to the speed it does it.
Though it does seem weird that the completed downloads fail, since there should be an integrity check when finished as well.

No it's not. No. There isn't. My internet connection is stable and solid.
And even if there were a problem that would not be a valid criticism. "There's something wrong with the road we used to deliver your closed package so we only delivered you half the package" is not a reasonable defense.
The transport medium is irrelevant here. The installer is able to tell that files are corrupted. I get a warning about the file. The only option shouldn't be "delete and start again". There should be a "reacquire this specific file" step automatically implemented. There is not. The default action for a link, button or interactable is to open it in the current tab.
This behaviour should only be changed where it cannot be avoided or doing so adds some functionality.
This is both form an accessibility and a UX point of view.
Changing it and offering nothing requiring an exception to part of what provides security to the internet is bad design, plain and simple.
Do popups or new tabs have a place? Yes.
Is this one of them? No.

You know you can turn popups off. I don't have any.

I assume he means the New Post popup.

That's a popup? I mean, that's considered a popup!? I'm lost.

Only case of a popup I can consider on GOG. Some others have encountered popups on GOG before, but turns out there was malware on their computer.

Okay, I better step back for a minute before I say something to offend the OP. :P I don't want to start off the new year alienating a GOG customer.

I get invalid downloads at times too. The best way is as JMich said, just clear all the games and ask it to redownload, or force the application closed and reopen - it then checks everything to see how far it got.
I have problems due to a very poor internet - speeds fluctuate wildly here these days 6bits/s at one point on Christmas day- and this can make downloads almost imposable some days. So I have to check everything I download has in fact downloaded correctly. Probably best not to assume anything with tinyE.

Yep, that's the method I use as well, although I rarely have a problem using the downloader. The only problems I have encountered somewhat frequently is when using the browser to download large files... using the Gog downloader is just more reliable in my experience.

Downloading a file essentially means copying it. I've been using computers for the past 23 years, and they have NEVER failed me when it came to copying data. Never was any result of copying an otherwise healthy file corrupt for me. The only time a copied file was corrupt was when the original file was corrupt in the first case. That's why I find it odd. Computers are too reliable when it comes to that.

So that's why I believe the GOG staff didn't include that option with the downloader. Unnecessary effort for something that happens very, very rarely. Or maybe the verifying software cannot verify files that haven't been downloaded in their entirety.

Wrong on both accounts.
The downloader does include a verify function, which works if you queue already downloaded files, or if you close and restart the downloader (though this part only if it remembers your queue). It also does the check by chunks, which for the games are 10MB each.

And faulty RAM can result in corrupted files, even though it's a rare case.

Not so weird, really. I only download my games through the GOG downloader and I did have corruption problems in the past, which I realised after I backed up my games. Ever since then, I always check the digital signature of all single-part installers that I download and always do an integrity check for multi-part ones, before copying it.